Let $g(x)=chi_B(x)$ be the indicator function of a bounded convex set in $Bbb R^d$, $dgeq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d neq 1 mod 4$, then there does not exist $S subset {Bbb R}^{2d}$ such that ${ {g(x-a)e^{2 pi i x cdot b} }}_{(a,b) in S}$ is an orthonormal basis for $L^2({Bbb R}^d)$.
设$g(x)=chi_B(x)$为$Bbb R^d$, $dgeq 2$中有界凸集的指示函数，该凸集边界光滑，处处具有不消失的高斯曲率。用组合方法证明了如果$d neq 1 mod 4$，则不存在$S subset {Bbb R}^{2d}$使得${ {g(x-a)e^{2 pi i x cdot b} }}_{(a,b) in S}$是$L^2({Bbb R}^d)$的正交基。
{"title":"Gabor orthogonal bases and convexity","authors":"A. Iosevich, A. Mayeli","doi":"10.19086/DA.5952","DOIUrl":"https://doi.org/10.19086/DA.5952","url":null,"abstract":"Let $g(x)=chi_B(x)$ be the indicator function of a bounded convex set in $Bbb R^d$, $dgeq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d neq 1 mod 4$, then there does not exist $S subset {Bbb R}^{2d}$ such that ${ {g(x-a)e^{2 pi i x cdot b} }}_{(a,b) in S}$ is an orthonormal basis for $L^2({Bbb R}^d)$.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46173330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Popular progression differences in vector spaces II, Discrete Analysis 2019:16, 39 pp. A central result in additive combinatorics, Roth's theorem, asserts that for every $delta>0$ there exists $n$ such that every set $Asubset{1,2,dots,n}$ of size at least $delta n$ contains an arithmetic progression of length 3. It was observed by Meshulam that Roth's argument can be easily adapted to prove a similar result in the vector space $mathbb F_p^n$. An averaging argument can then be used to prove that if ${x,x+d,x+2d}$ is an arithmetic progression of length 3 chosen at random in $mathbb F_p^n$, then the probability that it is a subset of $A$ is at least a positive constant $c(delta)$ that depends on $delta$ only. If $A$ is a randomly chosen set of density $delta$, then the probability above is approximately $delta^3$. Bergelson, Host and Kra noticed that although there are sets $A$ for which the probability is smaller than this, in all known examples there was at least one difference $d$ such that if $x$ is chosen randomly, then the probability that ${x,x+d,x+2d}subset A$ is at least $delta^3-o(1)$, so they conjectured that this was the case. This conjecture was proved by Green using an arithmetic analogue of Szemer'edi's regularity lemma that he had developed. However, the price that he had to pay for this strengthening of the vector space Roth theorem was that the dimension needed in order to guarantee the existence of such a "popular progression difference" $d$ was very high -- a tower of height a power of $epsilon^{-1}$ was needed to obtain a probability of $delta^3-epsilon$. This tower-type bound is typical of proofs that use regularity lemmas. While it is known that the regularity lemmas themselves actually require tower-type bounds, it is less clear whether that is true of their many _applications_, and indeed there are several examples of results that were initially proved with bad bounds using regularity lemmas and then later reproved with different arguments and much better bounds. A major open problem is to determine whether tower-type bounds are needed for the triangle removal lemma. (The best known bound for this result, a tower of logarithmic height, is due to the first-named author.) In a companion paper to this one, the authors proved that tower-type bounds are necessary for the theorem of Green mentioned above. More precisely, they obtained both an upper and lower bound for the dimension $n$, and both bounds were towers of height of order $log(1/epsilon)$. This was a very interesting contribution to the general question about applications of regularity lemmas, since it was the first time anybody had identified a non-trivial application of a regularity lemma for which it could be shown that a tower-type bound was necessary. (A possible definition of "non-trivial" here is a statement that does not easily imply a regularity lemma.) In this paper, the authors continue their investigation. They define $n_p(alpha,beta)$ to be the sm
{"title":"Popular progression differences in vector spaces II","authors":"J. Fox, H. Pham","doi":"10.19086/DA.11002","DOIUrl":"https://doi.org/10.19086/DA.11002","url":null,"abstract":"Popular progression differences in vector spaces II, Discrete Analysis 2019:16, 39 pp.\u0000\u0000A central result in additive combinatorics, Roth's theorem, asserts that for every $delta>0$ there exists $n$ such that every set $Asubset{1,2,dots,n}$ of size at least $delta n$ contains an arithmetic progression of length 3. It was observed by Meshulam that Roth's argument can be easily adapted to prove a similar result in the vector space $mathbb F_p^n$. An averaging argument can then be used to prove that if ${x,x+d,x+2d}$ is an arithmetic progression of length 3 chosen at random in $mathbb F_p^n$, then the probability that it is a subset of $A$ is at least a positive constant $c(delta)$ that depends on $delta$ only. \u0000\u0000If $A$ is a randomly chosen set of density $delta$, then the probability above is approximately $delta^3$. Bergelson, Host and Kra noticed that although there are sets $A$ for which the probability is smaller than this, in all known examples there was at least one difference $d$ such that if $x$ is chosen randomly, then the probability that ${x,x+d,x+2d}subset A$ is at least $delta^3-o(1)$, so they conjectured that this was the case. \u0000\u0000This conjecture was proved by Green using an arithmetic analogue of Szemer'edi's regularity lemma that he had developed. However, the price that he had to pay for this strengthening of the vector space Roth theorem was that the dimension needed in order to guarantee the existence of such a \"popular progression difference\" $d$ was very high -- a tower of height a power of $epsilon^{-1}$ was needed to obtain a probability of $delta^3-epsilon$. \u0000\u0000This tower-type bound is typical of proofs that use regularity lemmas. While it is known that the regularity lemmas themselves actually require tower-type bounds, it is less clear whether that is true of their many _applications_, and indeed there are several examples of results that were initially proved with bad bounds using regularity lemmas and then later reproved with different arguments and much better bounds. A major open problem is to determine whether tower-type bounds are needed for the triangle removal lemma. (The best known bound for this result, a tower of logarithmic height, is due to the first-named author.) \u0000\u0000In a companion paper to this one, the authors proved that tower-type bounds are necessary for the theorem of Green mentioned above. More precisely, they obtained both an upper and lower bound for the dimension $n$, and both bounds were towers of height of order $log(1/epsilon)$. This was a very interesting contribution to the general question about applications of regularity lemmas, since it was the first time anybody had identified a non-trivial application of a regularity lemma for which it could be shown that a tower-type bound was necessary. (A possible definition of \"non-trivial\" here is a statement that does not easily imply a regularity lemma.)\u0000\u0000In this paper, the authors continue their investigation. They define $n_p(alpha,beta)$ to be the sm","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49466641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
T. Anderson, Brian Cook, Kevin A. Hughes, A. Kumchev
Improved $ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions, Discrete Analysis 2018:10, 18pp. An important role in harmonic analysis is played by the notion of a _maximal function_ (which is actually a non-linear operator on a space of functions). The best-known example is the _Hardy-Littlewood maximal function_, which takes a function $f:mathbb R^dtomathbb C$ and replaces it by the function $Mf:mathbb R^dtomathbb R$, which is defined by the formula $$Mf(x)=sup_{r>0}frac 1{|B_r(x)|}int_{B_r(x)}|f(x)|dx,$$ where $B_r(x)$ is the ball of radius $r$ about $x$. In other words, $Mf$ is the largest average of $|f|$ over any ball centred at $x$. Particularly useful are inequalities bounding norms of $Mf$ in terms of norms of $f$: for example, it is known that if $1
{"title":"Improved ℓp -Boundedness for Integral k -Spherical Maximal Functions","authors":"T. Anderson, Brian Cook, Kevin A. Hughes, A. Kumchev","doi":"10.19086/da.3675","DOIUrl":"https://doi.org/10.19086/da.3675","url":null,"abstract":"Improved $ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions, Discrete Analysis 2018:10, 18pp.\u0000\u0000An important role in harmonic analysis is played by the notion of a _maximal function_ (which is actually a non-linear operator on a space of functions). The best-known example is the _Hardy-Littlewood maximal function_, which takes a function $f:mathbb R^dtomathbb C$ and replaces it by the function $Mf:mathbb R^dtomathbb R$, which is defined by the formula\u0000$$Mf(x)=sup_{r>0}frac 1{|B_r(x)|}int_{B_r(x)}|f(x)|dx,$$\u0000where $B_r(x)$ is the ball of radius $r$ about $x$. In other words, $Mf$ is the largest average of $|f|$ over any ball centred at $x$. Particularly useful are inequalities bounding norms of $Mf$ in terms of norms of $f$: for example, it is known that if $1<pleqinfty$, then there is a constant $A_{p,d}$ such that $|Mf|_pleq A_{p,d}|f|_p$ for every function $fin L_p(mathbb R^d)$.\u0000\u0000This paper concerns discrete maximal functions, where $f$ is now defined on $mathbb Z^d$. They are also _spherical_ maximal functions, meaning that the averages are over spheres rather than balls. And finally, the spheres are not (necessarily) Euclidean spheres but spheres in $ell_k^d$ for more general $k$ than just 2. To be precise, if $k$ is a positive integer, $lambda$ is a positive real, and $x$ is a point in $mathbb Z^d$, define $S(x,k,lambda)$ to be the set ${yinmathbb Z^d:sum_{i=1}^d|x_i-y_i|^k=lambda}$. The authors look at the corresponding maximal function, which takes $f$ to the function $Mf$ defined by\u0000$$Mf(x)=mathop{sup}_{S(x,k,lambda)neemptyset}frac 1{|S(x,k,lambda)|}sum_{yin S(x,k,lambda)}|f(y)|.$$\u0000Their object is to understand, given $d$ and $k$, for which $p$ this maximal function is bounded: that is, for which $p$ there exists a constant $A_{p,k,d}$ such that $|Mf|_pleq A_{p,d,k}|f|_p$, where $|.|_p$ is the $ell_p$ norm for functions defined on $mathbb Z^d$. It turns out that the larger the value of $p$, the easier it is for a maximal function to be bounded (an indication that this is to be expected is that it is trivially bounded when $p=infty$), so the aim is to prove boundedness with $p$ as small as possible. The authors manage to prove boundedness when $p$ is at least a certain value $p_0(d,k)$, which is given by a slightly complicated expression that can be found on the second page of their paper. However, the key point is that $p_0(d,k)<2$ when $d$ is sufficiently large in terms of $k$, and the \"sufficiently large\" they require is only quadratic, whereas the previous best known results required $d$ to be at least cubic in $k$.\u0000\u0000Not surprisingly, there are connections between this question and Waring's problem, since both involve writing an integer as a sum of a certain number of $k$th powers. In particular, if $d$ is sufficiently large compared with $k$, good estimates are available for the sizes of the spheres $S(x,k,lambda)$, which allow one to replace the maximal function given above by a simpler one that is equivalent (in the sen","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44759893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A geometric simulation theorem on direct products of finitely generated groups, Discrete Analysis 2019:9, 25 pp. In 1961, the mathematician and philosopher Hao Wang introduced the notion that we now call a _Wang tiling_ of $mathbb Z^2$. Each tile is a unit square with edges marked in a certain way, and two tiles can be placed next to each other if and only if the adjacent edges have the same marking. (Also, tiles are not allowed to be rotated.) Wang observed that if every set of Wang tiles admits a periodic tiling, then there is an algorithm for deciding whether any given finite set of Wang tiles can tile the plane, since if it cannot, then by an easy compactness argument there is some finite subset of the plane that cannot be tiled, whereas if it can, then one can do a brute-force search for a fundamental domain of the tiling. Wang believed that every tiling should be periodic, but his student Robert Berger showed in his PhD thesis in 1964 that the the tiling problem could be used to encode the halting problem, thereby proving that there was no algorithm for solving the former, which yielded the first known example of a set of tiles that could tile the plane but only aperiodically. (The result appeared in paper form in 1966.) The notions of Wang tilings and aperiodicity have been fruitfully generalized from $mathbb Z^2$ to more general groups, as follows. Let $G$ be a group (which will be infinite) and let $mathcal A$ be a set of symbols, which we can think of as our "tiles". Define a _pattern_ to be a function $p:Xtomathcal A$, where $X$ is some finite subset of $G$. Given a function $omega:Gtomathcal A$, say that the pattern $p$ _occurs in_ $omega$ if there exists some $gin G$ such that $omega(gx)=p(x)$ for every $xin X$. Now let $P$ be a finite set of forbidden patterns. The set of all functions $omega$ such that no pattern $pin P$ occurs in $omega$ is called the _subshift of finite type_ determined by $P$. The functions $omega$ that satisfy this condition are called _configurations_. Note that the set of tilings of the plane with a given set of Wang tiles is a subshift of finite type determined by the rules that stipulate when two tiles may be placed next to each other, each of which is given by a forbidden pattern of size 2. Note that $G$ acts on a subshift of finite type in an obvious way: if $hin G$ and $omega$ is a configuration, then so is the function $omega_h$ defined by $omega_h(g)=omega(h^{-1}g)$. The map $(h,omega)mapstoomega_h$ is easily checked to be an action. The subshift is called _strongly aperiodic_ if this action is free: that is, if no non-trivial group element takes any configuration to itself. In the case of Wang tiles, the subshift is strongly aperiodic if and only if no tiling with the given set of tiles is periodic. This paper is about so-called _simulation theorems_. A basic example of such a theorem is the following result mentioned by the authors. A group is said to be _recursively presented_ if it is counta
{"title":"A geometric simulation theorem on direct products of finitely generated groups","authors":"S. Barbieri","doi":"10.19086/da.8820","DOIUrl":"https://doi.org/10.19086/da.8820","url":null,"abstract":"A geometric simulation theorem on direct products of finitely generated groups, Discrete Analysis 2019:9, 25 pp.\u0000\u0000In 1961, the mathematician and philosopher Hao Wang introduced the notion that we now call a _Wang tiling_ of $mathbb Z^2$. Each tile is a unit square with edges marked in a certain way, and two tiles can be placed next to each other if and only if the adjacent edges have the same marking. (Also, tiles are not allowed to be rotated.) \u0000\u0000Wang observed that if every set of Wang tiles admits a periodic tiling, then there is an algorithm for deciding whether any given finite set of Wang tiles can tile the plane, since if it cannot, then by an easy compactness argument there is some finite subset of the plane that cannot be tiled, whereas if it can, then one can do a brute-force search for a fundamental domain of the tiling. Wang believed that every tiling should be periodic, but his student Robert Berger showed in his PhD thesis in 1964 that the the tiling problem could be used to encode the halting problem, thereby proving that there was no algorithm for solving the former, which yielded the first known example of a set of tiles that could tile the plane but only aperiodically. (The result appeared in paper form in 1966.)\u0000\u0000The notions of Wang tilings and aperiodicity have been fruitfully generalized from $mathbb Z^2$ to more general groups, as follows. Let $G$ be a group (which will be infinite) and let $mathcal A$ be a set of symbols, which we can think of as our \"tiles\". Define a _pattern_ to be a function $p:Xtomathcal A$, where $X$ is some finite subset of $G$. Given a function $omega:Gtomathcal A$, say that the pattern $p$ _occurs in_ $omega$ if there exists some $gin G$ such that $omega(gx)=p(x)$ for every $xin X$. Now let $P$ be a finite set of forbidden patterns. The set of all functions $omega$ such that no pattern $pin P$ occurs in $omega$ is called the _subshift of finite type_ determined by $P$. The functions $omega$ that satisfy this condition are called _configurations_. Note that the set of tilings of the plane with a given set of Wang tiles is a subshift of finite type determined by the rules that stipulate when two tiles may be placed next to each other, each of which is given by a forbidden pattern of size 2. \u0000\u0000Note that $G$ acts on a subshift of finite type in an obvious way: if $hin G$ and $omega$ is a configuration, then so is the function $omega_h$ defined by $omega_h(g)=omega(h^{-1}g)$. The map $(h,omega)mapstoomega_h$ is easily checked to be an action. The subshift is called _strongly aperiodic_ if this action is free: that is, if no non-trivial group element takes any configuration to itself. In the case of Wang tiles, the subshift is strongly aperiodic if and only if no tiling with the given set of tiles is periodic.\u0000\u0000This paper is about so-called _simulation theorems_. A basic example of such a theorem is the following result mentioned by the authors. A group is said to be _recursively presented_ if it is counta","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43261970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There exists an absolute constant $delta > 0$ such that for all $q$ and all subsets $A subseteq mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - delta}$, then [ |(A-A)(A-A)| = |{ (a -b) (c-d) : a,b,c,d in A}| > frac{q}{2}. ] Any $delta q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions. Our proof is based on a qualitatively optimal characterisation of sets $A,X subseteq mathbb{F}_q$ for which the number of solutions to the equation [ (a_1-a_2) = x (a_3-a_4) , , ; a_1,a_2, a_3, a_4 in A, x in X ] is nearly maximum. A key ingredient is determining exact algebraic structure of sets $A, X$ for which $|A + XA|$ is nearly minimum, which refines a result of Bourgain and Glibichuk using work of Gill, Helfgott, and Tao. We also prove a stronger statement for [ (A-B)(C-D) = { (a -b) (c-d) : a in A, b in B, c in C, d in D} ] when $A,B,C,D$ are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.
存在一个绝对常数$delta > 0$，使得对于所有$q$和所有具有$q$元素的有限域的子集$A subseteq mathbb{F}_q$，如果$|A| > q^{2/3 - delta}$，那么[ |(A-A)(A-A)| = |{ (a -b) (c-d) : a,b,c,d in A}| > frac{q}{2}. ] Any $delta q^{2/3}$，由于Bennett, Hart, Iosevich, Pakianathan和Rudnev，这是此类问题的典型。我们的证明是基于集合$A,X subseteq mathbb{F}_q$的定性最优特征，其中方程[ (a_1-a_2) = x (a_3-a_4) , , ; a_1,a_2, a_3, a_4 in A, x in X ]的解的数量几乎是最大的。一个关键因素是确定集合$A, X$的精确代数结构，其中$|A + XA|$几乎是最小的，这改进了Bourgain和Glibichuk使用Gill, Helfgott和Tao的工作的结果。当$A,B,C,D$是素域上的集合时，我们也证明了一个更强的命题[ (A-B)(C-D) = { (a -b) (c-d) : a in A, b in B, c in C, d in D} ]，推广了Roche-Newton, Rudnev, Shkredov和作者的结果。
{"title":"Products of Differences over Arbitrary Finite Fields","authors":"B. Murphy, G. Petridis","doi":"10.19086/DA.5098","DOIUrl":"https://doi.org/10.19086/DA.5098","url":null,"abstract":"There exists an absolute constant $delta > 0$ such that for all $q$ and all subsets $A subseteq mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - delta}$, then [ |(A-A)(A-A)| = |{ (a -b) (c-d) : a,b,c,d in A}| > frac{q}{2}. ] Any $delta q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions. \u0000Our proof is based on a qualitatively optimal characterisation of sets $A,X subseteq mathbb{F}_q$ for which the number of solutions to the equation [ (a_1-a_2) = x (a_3-a_4) , , ; a_1,a_2, a_3, a_4 in A, x in X ] is nearly maximum. \u0000A key ingredient is determining exact algebraic structure of sets $A, X$ for which $|A + XA|$ is nearly minimum, which refines a result of Bourgain and Glibichuk using work of Gill, Helfgott, and Tao. \u0000We also prove a stronger statement for [ (A-B)(C-D) = { (a -b) (c-d) : a in A, b in B, c in C, d in D} ] when $A,B,C,D$ are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43650789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Good bounds in certain systems of true complexity one, Discrete Analysis 2018:21, 40 pp. In his analytic proof of Szemeredi's theorem, Gowers introduced a sequence of norms, now known as the $U^k$ norms, given by the formula $$|f|_{U^k}^{2^k}=mathbb E_{x,a_1,dots,a_k}prod_{epsilonin{0,1}^k}C^{|epsilon|}fBigl(x+sum_iepsilon_ia_iBigr),$$ where $f$ is a complex-valued function defined on a finite Abelian group, $C$ is the operation of complex conjugation and $|epsilon|=sum_iepsilon_i$. For example, $$|f|_{U^2}^4=mathbb E_{x,a,b}f(x)overline{f(x+a)f(x+b)}f(x+a+b).$$ A key lemma in his proof was that for any $k$ such functions $f_1,dots,f_k$ that take values of modulus at most 1, one has the inequality $$|mathbb E_{x,d}f_1(x)f_2(x+d)dots f_k(x+(k-1)d)|leqmin_i|f_i|_{U^{k-1}}.$$ This lemma was proved using repeated applications of the Cauchy-Schwarz inequality. In such a situation we say that the average on the left-hand side is _controlled by the $U^{k-1}$ norm_. Later, as part of their work on linear equations in the primes, Green and Tao considered more general expressions such as $$mathbb E_{x,y,z}f(x+y)f(y+z)f(x+z)f(x+y+z),$$ finding for each one a $k$ such that the expression is controlled by the $U^k$ norm. Their proof was a generalization of that of Gowers -- again, it used repeated applications of the Cauchy-Schwarz inequality. The $U^k$ norms increase with $k$, so if the average is controlled by the $U^k$ norm, then it is controlled by the $U^ell$ norm for all $ell>k$. It is therefore natural to ask what the minimal $k$ is that works for any given average of a product over linear forms. This question was examined in a series of papers of Gowers and Wolf, who conjectured that the average $$mathbb E_{x_1,dots,x_r}f_1(L_1(x_1,dots,x_r)dots f_s(L_s(x_1,dots,x_r))$$ is controlled by the $U^k$ norm if and only if the functions $L_1^k,dots,L_s^k$ are linearly independent. To understand what this says, note that if the linear forms are $x, x+d, x+2d, x+3d$, then we have that $x^2-3(x+d)^2+3(x+2d)^2-(x+3d)^2=0$, so the $U^2$ norm does not control the average $mathbb E_{x,d}f(x)f(x+d)f(x+2d)f(x+3d)$, but the cubes of the functions are linearly independent, so the $U^3$ norm does control it. Simple examples show that the independence condition is necessary, and in the case of arithmetic progressions, it agrees with the bound in the other direction that comes from the Cauchy-Schwarz argument. However, there are other systems of linear forms where the conjectured bound is strictly smaller than the bound proved by Green and Tao. Gowers and Wolf proved their conjecture in many cases, and the remaining cases were proved by Green and Tao, and Hatami, Hatami and Lovett. All these proofs used difficult results known as inverse theorems for the $U^k$ norms, due to Bergelson, Tao and Ziegler, and Green, Tao and Ziegler. Since those theorems were known with only very poor bounds, the dependence of the averages on the $U^k$ norm given by these proo
{"title":"Good Bounds in Certain Systems of True Complexity One","authors":"Freddie Manners","doi":"10.19086/da.6814","DOIUrl":"https://doi.org/10.19086/da.6814","url":null,"abstract":"Good bounds in certain systems of true complexity one, Discrete Analysis 2018:21, 40 pp.\u0000\u0000In his analytic proof of Szemeredi's theorem, Gowers introduced a sequence of norms, now known as the $U^k$ norms, given by the formula\u0000\u0000$$|f|_{U^k}^{2^k}=mathbb E_{x,a_1,dots,a_k}prod_{epsilonin{0,1}^k}C^{|epsilon|}fBigl(x+sum_iepsilon_ia_iBigr),$$\u0000\u0000where $f$ is a complex-valued function defined on a finite Abelian group, $C$ is the operation of complex conjugation and $|epsilon|=sum_iepsilon_i$. For example, \u0000\u0000$$|f|_{U^2}^4=mathbb E_{x,a,b}f(x)overline{f(x+a)f(x+b)}f(x+a+b).$$\u0000\u0000A key lemma in his proof was that for any $k$ such functions $f_1,dots,f_k$ that take values of modulus at most 1, one has the inequality\u0000\u0000$$|mathbb E_{x,d}f_1(x)f_2(x+d)dots f_k(x+(k-1)d)|leqmin_i|f_i|_{U^{k-1}}.$$\u0000\u0000This lemma was proved using repeated applications of the Cauchy-Schwarz inequality. In such a situation we say that the average on the left-hand side is _controlled by the $U^{k-1}$ norm_. \u0000\u0000Later, as part of their work on linear equations in the primes, Green and Tao considered more general expressions such as\u0000\u0000$$mathbb E_{x,y,z}f(x+y)f(y+z)f(x+z)f(x+y+z),$$\u0000\u0000finding for each one a $k$ such that the expression is controlled by the $U^k$ norm. Their proof was a generalization of that of Gowers -- again, it used repeated applications of the Cauchy-Schwarz inequality. \u0000\u0000The $U^k$ norms increase with $k$, so if the average is controlled by the $U^k$ norm, then it is controlled by the $U^ell$ norm for all $ell>k$. It is therefore natural to ask what the minimal $k$ is that works for any given average of a product over linear forms. This question was examined in a series of papers of Gowers and Wolf, who conjectured that the average\u0000\u0000$$mathbb E_{x_1,dots,x_r}f_1(L_1(x_1,dots,x_r)dots f_s(L_s(x_1,dots,x_r))$$\u0000\u0000is controlled by the $U^k$ norm if and only if the functions $L_1^k,dots,L_s^k$ are linearly independent. To understand what this says, note that if the linear forms are $x, x+d, x+2d, x+3d$, then we have that $x^2-3(x+d)^2+3(x+2d)^2-(x+3d)^2=0$, so the $U^2$ norm does not control the average $mathbb E_{x,d}f(x)f(x+d)f(x+2d)f(x+3d)$, but the cubes of the functions are linearly independent, so the $U^3$ norm does control it. Simple examples show that the independence condition is necessary, and in the case of arithmetic progressions, it agrees with the bound in the other direction that comes from the Cauchy-Schwarz argument. However, there are other systems of linear forms where the conjectured bound is strictly smaller than the bound proved by Green and Tao.\u0000\u0000Gowers and Wolf proved their conjecture in many cases, and the remaining cases were proved by Green and Tao, and Hatami, Hatami and Lovett. All these proofs used difficult results known as inverse theorems for the $U^k$ norms, due to Bergelson, Tao and Ziegler, and Green, Tao and Ziegler. Since those theorems were known with only very poor bounds, the dependence of the averages on the $U^k$ norm given by these proo","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44258211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On the number of points in general position in the plane, Discrete Analysis 2018:16, 20 pp. A recurring theme in combinatorics is questions of the following kind. Suppose that we have a combinatorial structure $S$ of size $n$ that contains no object of type $A$. Then how large a subset of $S$ can we find that contains no object of type $B$? For example, a graph with $n$ vertices that contains no clique of size 4 can be shown quite easily to have a triangle-free subgraph with $n^{1/2}$ vertices, and Wolfovitz has shown that there are graphs with no clique of size 4 and no triangle-free subgraph with more than $n^{1/2}(log n)^{120}$ vertices. One of the main questions discussed in this paper is perhaps the first question of this type one would think of in discrete geometry: if $S$ is a set of $n$ points in the plane and if no four of these points are collinear, then how large a subset of $S$ can one find with no three collinear? A bound of $o(n)$ follows from the density Hales-Jewett theorem, which implies that a subset of ${1,2,3}^k$ of positive density contains three points in a line. It is not hard to project the set ${1,2,3}$ into the plane in such a way that collinearity is preserved, but no four points of the image lie in a line. However, the bound obtained this way is very weak -- roughly $n/log_*(n)$. This paper obtains the first reasonable bound for the problem, namely $n^{5/6+o(1)}$. It is not clear whether 5/6 is the right exponent, but the authors suggest that their construction may be close to optimal and that the difficulty is to calculate the correct exponent for that example. Perhaps the most interesting aspect of the paper is that it uses the so-called method of containers. This method, developed by Saxton and Thomason, and independently by Balogh, Morris and Samotij, has already been used to solve a large number of important problems, but this appears to be the first time it has been used to solve a problem in discrete geometry, and it is used in a novel way. They also use containers to prove a second discrete geometry result, this time about epsilon-nets. Given a family $mathcal F$ of subsets of a finite set $X$, an $epsilon$-net $E$ of $mathcal F$ is a subset $E$ of $X$ such that every $Finmathcal F$ of size at least $epsilon|X|$ contains an element of $E$. There are many interesting questions about the sizes of $epsilon$-nets when $X$ is a geometrical set such as a finite set of points in the plane, and $mathcal F$ is some natural class of subsets such as the set of all intersections of $X$ with convex bodies. With this example, one can also define a _weak_ $epsilon$-net as follows: it is a set of points $E$ in the plane, not necessarily a subset of $X$, such that every convex hull of at least $epsilon|X|$ points of $X$ contains a point of $E$. Natural notions of weak $epsilon$-nets can be defined in many other contexts too. An interesting open question, asked by Noga Alon, is whether there is some natural geometrically
{"title":"On the number of points in general position in the plane","authors":"J. Balogh, J. Solymosi","doi":"10.19086/da.4438","DOIUrl":"https://doi.org/10.19086/da.4438","url":null,"abstract":"On the number of points in general position in the plane, Discrete Analysis 2018:16, 20 pp.\u0000\u0000A recurring theme in combinatorics is questions of the following kind. Suppose that we have a combinatorial structure $S$ of size $n$ that contains no object of type $A$. Then how large a subset of $S$ can we find that contains no object of type $B$? For example, a graph with $n$ vertices that contains no clique of size 4 can be shown quite easily to have a triangle-free subgraph with $n^{1/2}$ vertices, and Wolfovitz has shown that there are graphs with no clique of size 4 and no triangle-free subgraph with more than $n^{1/2}(log n)^{120}$ vertices. \u0000\u0000One of the main questions discussed in this paper is perhaps the first question of this type one would think of in discrete geometry: if $S$ is a set of $n$ points in the plane and if no four of these points are collinear, then how large a subset of $S$ can one find with no three collinear? \u0000\u0000A bound of $o(n)$ follows from the density Hales-Jewett theorem, which implies that a subset of ${1,2,3}^k$ of positive density contains three points in a line. It is not hard to project the set ${1,2,3}$ into the plane in such a way that collinearity is preserved, but no four points of the image lie in a line. However, the bound obtained this way is very weak -- roughly $n/log_*(n)$. This paper obtains the first reasonable bound for the problem, namely $n^{5/6+o(1)}$. It is not clear whether 5/6 is the right exponent, but the authors suggest that their construction may be close to optimal and that the difficulty is to calculate the correct exponent for that example.\u0000\u0000Perhaps the most interesting aspect of the paper is that it uses the so-called method of containers. This method, developed by Saxton and Thomason, and independently by Balogh, Morris and Samotij, has already been used to solve a large number of important problems, but this appears to be the first time it has been used to solve a problem in discrete geometry, and it is used in a novel way.\u0000\u0000They also use containers to prove a second discrete geometry result, this time about epsilon-nets. Given a family $mathcal F$ of subsets of a finite set $X$, an $epsilon$-net $E$ of $mathcal F$ is a subset $E$ of $X$ such that every $Finmathcal F$ of size at least $epsilon|X|$ contains an element of $E$. There are many interesting questions about the sizes of $epsilon$-nets when $X$ is a geometrical set such as a finite set of points in the plane, and $mathcal F$ is some natural class of subsets such as the set of all intersections of $X$ with convex bodies. With this example, one can also define a _weak_ $epsilon$-net as follows: it is a set of points $E$ in the plane, not necessarily a subset of $X$, such that every convex hull of at least $epsilon|X|$ points of $X$ contains a point of $E$. Natural notions of weak $epsilon$-nets can be defined in many other contexts too.\u0000\u0000An interesting open question, asked by Noga Alon, is whether there is some natural geometrically","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49149042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the matrix $A_2$ conjecture for the dyadic square function, that is, a norm estimate of the matrix weighted square function, where the focus is on the sharp linear dependence on the matrix $A_2$ constant in the estimate. Moreover, we give a mixed estimate in terms of $A_2$ and $A_{infty}$ constants. Key is a sparse domination of a process inspired by the integrated form of the matrix--weighted square function.
{"title":"The sharp square function estimate with matrix weight","authors":"T. Hytonen, S. Petermichl, A. Volberg","doi":"10.19086/da.7597","DOIUrl":"https://doi.org/10.19086/da.7597","url":null,"abstract":"We prove the matrix $A_2$ conjecture for the dyadic square function, that is, a norm estimate of the matrix weighted square function, where the focus is on the sharp linear dependence on the matrix $A_2$ constant in the estimate. Moreover, we give a mixed estimate in terms of $A_2$ and $A_{infty}$ constants. Key is a sparse domination of a process inspired by the integrated form of the matrix--weighted square function.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41375804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of ${0,1}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on ${0,1}^n$. We show that for any $m$-element subset $mathcal{F} subset {0,1}^n$ and any integer $l$, if the edge boundary of $mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $mathcal{G} subset {0,1}^n$ such that $|mathcal{F} Delta mathcal{G}| leq Cl$, where $C$ is an absolute constant. This is best-possible, up to the value of $C$. Our result can be seen as a `stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space.
{"title":"On the structure of subsets of the discrete cube with small edge boundary","authors":"David Ellis, Nathan Keller, Noam Lifshitz","doi":"10.19086/DA.3668","DOIUrl":"https://doi.org/10.19086/DA.3668","url":null,"abstract":"The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of ${0,1}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on ${0,1}^n$. We show that for any $m$-element subset $mathcal{F} subset {0,1}^n$ and any integer $l$, if the edge boundary of $mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $mathcal{G} subset {0,1}^n$ such that $|mathcal{F} Delta mathcal{G}| leq Cl$, where $C$ is an absolute constant. This is best-possible, up to the value of $C$. Our result can be seen as a `stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2016-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68393030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of Bilu's result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a version of Bilu's geometry-of-numbers argument carried out in a nilpotent Lie algebra. We also present some applications. We verify a conjecture of Benjamini that if $S$ is a symmetric generating set for a group such that $1in S$ and $|S^n|le Mn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growth of the same degree $D$ at all subsequent scales, in the sense that $|S^r|ll_{M,D}r^D$ for every $rge n$. Our methods also provide an important ingredient in a forthcoming companion paper in which we show that if $(Gamma_n,S_n)$ is a sequence of Cayley graphs satisfying $|S_n^n|ll n^D$ as $ntoinfty$, and if $m_ngg n$ as $ntoinfty$, then every Gromov-Hausdorff limit of the sequence $(Gamma_{n},frac{d_{S_{n}}}{m_n})$ has homogeneous dimension bounded by $D$. We also note that our arguments imply that every approximate group has a large subset with a large quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent group of bounded rank and step.
{"title":"Properness of nilprogressions and the persistence of polynomial growth of given degree","authors":"R. Tessera, Matthew C. H. Tointon","doi":"10.19086/DA.5056","DOIUrl":"https://doi.org/10.19086/DA.5056","url":null,"abstract":"We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of Bilu's result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a version of Bilu's geometry-of-numbers argument carried out in a nilpotent Lie algebra. We also present some applications. We verify a conjecture of Benjamini that if $S$ is a symmetric generating set for a group such that $1in S$ and $|S^n|le Mn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growth of the same degree $D$ at all subsequent scales, in the sense that $|S^r|ll_{M,D}r^D$ for every $rge n$. Our methods also provide an important ingredient in a forthcoming companion paper in which we show that if $(Gamma_n,S_n)$ is a sequence of Cayley graphs satisfying $|S_n^n|ll n^D$ as $ntoinfty$, and if $m_ngg n$ as $ntoinfty$, then every Gromov-Hausdorff limit of the sequence $(Gamma_{n},frac{d_{S_{n}}}{m_n})$ has homogeneous dimension bounded by $D$. We also note that our arguments imply that every approximate group has a large subset with a large quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent group of bounded rank and step.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2016-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68392763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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