We prove that on cyclic groups of square-free order, a tile is a spectral set. Moreover, we prove that the converse also holds on cyclic groups $mathbb{Z}_{pqr}$ with $p,q,r$ distinct primes, that is to say, a spectral set is also a tile. As a consequence, Fuglede's conjecture holds on cyclic groups $mathbb{Z}_{pqr}$.
{"title":"Fuglede's conjecture holds on cyclic groups $mathbb{Z}_{pqr}$","authors":"Ruxi Shi","doi":"10.19086/da.10570","DOIUrl":"https://doi.org/10.19086/da.10570","url":null,"abstract":"We prove that on cyclic groups of square-free order, a tile is a spectral set. Moreover, we prove that the converse also holds on cyclic groups $mathbb{Z}_{pqr}$ with $p,q,r$ distinct primes, that is to say, a spectral set is also a tile. As a consequence, Fuglede's conjecture holds on cyclic groups $mathbb{Z}_{pqr}$.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45664643","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}
{"title":"Exponential sums with reducible polynomials","authors":"C. Dartyge, G. Martin","doi":"10.19086/da.10793","DOIUrl":"https://doi.org/10.19086/da.10793","url":null,"abstract":"Hooley proved that if $fin Bbb Z [X]$ is irreducible of degree $ge 2$, then the fractions ${ r/n}$, $0<r<n$ with $f(r)equiv 0pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible polynomials of degree $le 3$. In particular, we establish asymptotic formulas for exponential sums over these normalized roots.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42404708","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}
Efficient arithmetic regularity and removal lemmas for induced bipartite patterns, Discrete Analysis 2019:3, 14 pp. This paper provides a common extension of two recent lines of work: the study of arithmetic regularity lemmas under the model-theoretic assumption of stability initiated by Terry and Wolf, and that of graph regularity lemmas for graphs of bounded VC-dimension provided by Lovasz and Szegedy (following prior work of Alon, Fischer and Newman) and extended to hypergraphs by Fox, Pach and Suk. Since Szemeredi’s seminal work in the 1970s, regularity lemmas have proved to be of fundamental importance in many areas of discrete mathematics. In the graph setting, a regularity lemma states that the vertex set of any sufficiently large graph can be partitioned into a bounded number of sets such that almost all pairs of parts from the partition induce a bipartite graph that looks a lot like a random graph (that is, it is a quasi-random graph, in a sense that can be made precise in several essentially equivalent ways). An arithmetic analogue of Szemeredi's regularity lemma was formulated and proved by Green in 2005. An important special case of Green's lemma asserts that for any sufficiently large $n$ and any subset $A$ of the vector space $mathbb{F}_p^n$, this space can be partitioned into cosets of a subspace $H$ of bounded codimension such that the set $A$ behaves quasi-randomly with respect to almost every coset in the partition. (Here the quasi-random behaviour is defined in terms of the absolute value of the Fourier transform of the indicator function of the set $A$ relative to the subspace $H$.) In both settings, it was shown (by Gowers and Green, respectively) that the trade-off between the number of parts in the partition and the degree of quasi-randomness obtained was necessarily of tower-type. In the case of graphs, it had already been observed many years earlier that the existence of "irregular" pairs in the partition could not in general be excluded. That is, in general, the conclusions of the regularity lemma cannot be strengthened in either setting. The folklore example ruling out the existence of a completely regular graph partition is the _half-graph_, which is a bipartite graph defined on two vertex classes $X={x_1,x_2,dots,x_k}$ and $Y={y_1, y_2,dots, y_k}$, with edges between $x_i$ and $y_j$ if and only if $ileq j$. Malliaris and Shelah observed in 2014 that by forbidding induced copies of the half-graph (of constant size), one can indeed guarantee a completely regular partition of any sufficiently large graph. In fact, they proved an even stronger result: the number of parts of the partition depends polynomially on the regularity parameter, and the edge density between any two parts of the partition is guaranteed to be either close to 0 or close to 1. The half-graph is known to model theorists as a particular instance of the so-called "order property" (in this case, it is a property of the formula defining the edge r
{"title":"Efficient arithmetic regularity and removal lemmas for induced bipartite patterns","authors":"N. Alon, J. Fox, Yufei Zhao","doi":"10.19086/da.7757","DOIUrl":"https://doi.org/10.19086/da.7757","url":null,"abstract":"Efficient arithmetic regularity and removal lemmas for induced bipartite patterns, Discrete Analysis 2019:3, 14 pp.\u0000\u0000This paper provides a common extension of two recent lines of work: the study of arithmetic regularity lemmas under the model-theoretic assumption of stability initiated by Terry and Wolf, and that of graph regularity lemmas for graphs of bounded VC-dimension provided by Lovasz and Szegedy (following prior work of Alon, Fischer and Newman) and extended to hypergraphs by Fox, Pach and Suk. \u0000\u0000Since Szemeredi’s seminal work in the 1970s, regularity lemmas have proved to be of fundamental importance in many areas of discrete mathematics. In the graph setting, a regularity lemma states that the vertex set of any sufficiently large graph can be partitioned into a bounded number of sets such that almost all pairs of parts from the partition induce a bipartite graph that looks a lot like a random graph (that is, it is a quasi-random graph, in a sense that can be made precise in several essentially equivalent ways). \u0000\u0000An arithmetic analogue of Szemeredi's regularity lemma was formulated and proved by Green in 2005. An important special case of Green's lemma asserts that for any sufficiently large $n$ and any subset $A$ of the vector space $mathbb{F}_p^n$, this space can be partitioned into cosets of a subspace $H$ of bounded codimension such that the set $A$ behaves quasi-randomly with respect to almost every coset in the partition. (Here the quasi-random behaviour is defined in terms of the absolute value of the Fourier transform of the indicator function of the set $A$ relative to the subspace $H$.)\u0000\u0000In both settings, it was shown (by Gowers and Green, respectively) that the trade-off between the number of parts in the partition and the degree of quasi-randomness obtained was necessarily of tower-type. In the case of graphs, it had already been observed many years earlier that the existence of \"irregular\" pairs in the partition could not in general be excluded. That is, in general, the conclusions of the regularity lemma cannot be strengthened in either setting.\u0000\u0000The folklore example ruling out the existence of a completely regular graph partition is the _half-graph_, which is a bipartite graph defined on two vertex classes $X={x_1,x_2,dots,x_k}$ and $Y={y_1, y_2,dots, y_k}$, with edges between $x_i$ and $y_j$ if and only if $ileq j$. Malliaris and Shelah observed in 2014 that by forbidding induced copies of the half-graph (of constant size), one can indeed guarantee a completely regular partition of any sufficiently large graph. In fact, they proved an even stronger result: the number of parts of the partition depends polynomially on the regularity parameter, and the edge density between any two parts of the partition is guaranteed to be either close to 0 or close to 1. \u0000\u0000The half-graph is known to model theorists as a particular instance of the so-called \"order property\" (in this case, it is a property of the formula defining the edge r","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48853164","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 Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.
{"title":"The Fourier restriction and Kakeya problems over rings of integers modulo N","authors":"J. Hickman, James Wright","doi":"10.19086/DA.3682","DOIUrl":"https://doi.org/10.19086/DA.3682","url":null,"abstract":"The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46247671","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 give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms $Q$.
{"title":"Counting rational points on quadric surfaces","authors":"T. Browning, Roger Heath-Brown","doi":"10.19086/da.4375","DOIUrl":"https://doi.org/10.19086/da.4375","url":null,"abstract":"We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms $Q$.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44945708","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 halfspace is a function $fcolon{-1,1}^n rightarrow {0,1}$ of the form $f(x)=mathbb{1}(acdot x>t)$, where $sum_i a_i^2=1$. We show that if $f$ is a halfspace with $mathbb{E}[f]=epsilon$ and $a'=max_i |a_i|$, then the degree-1 Fourier weight of $f$ is $W^1(f)=Theta(epsilon^2 log(1/epsilon))$, and the maximal influence of $f$ is $I_{max}(f)=Theta(epsilon min(1,a' sqrt{log(1/epsilon)}))$. These results, which determine the exact asymptotic order of $W^1(f)$ and $I_{max}(f)$, provide sharp generalizations of theorems proved by Matulef, O'Donnell, Rubinfeld, and Servedio, and settle a conjecture posed by Kalai, Keller and Mossel. In addition, we present a refinement of the definition of noise sensitivity which takes into consideration the bias of the function, and show that (like in the unbiased case) halfspaces are noise resistant, and, in the other direction, any noise resistant function is well correlated with a halfspace. Our main tools are 'local' forms of the classical Chernoff inequality, like the following one proved by Devroye and Lugosi (2008): Let ${ x_i }$ be independent random variables uniformly distributed in ${-1,1}$, and let $a_iinmathbb{R}_+$ be such that $sum_i a_{i}^{2}=1$. If for some $tgeq 0$ we have $Pr[sum_{i} a_i x_i > t]=epsilon$, then $Pr[sum_{i} a_i x_i>t+delta]leq frac{epsilon}{2}$ holds for $deltaleq c/sqrt{log(1/epsilon)}$, where $c$ is a universal constant.
{"title":"Biased halfspaces, noise sensitivity, and local Chernoff inequalities","authors":"Nathan Keller, Ohad Klein","doi":"10.19086/DA.10234","DOIUrl":"https://doi.org/10.19086/DA.10234","url":null,"abstract":"A halfspace is a function $fcolon{-1,1}^n rightarrow {0,1}$ of the form $f(x)=mathbb{1}(acdot x>t)$, where $sum_i a_i^2=1$. \u0000We show that if $f$ is a halfspace with $mathbb{E}[f]=epsilon$ and $a'=max_i |a_i|$, then the degree-1 Fourier weight of $f$ is \u0000$W^1(f)=Theta(epsilon^2 log(1/epsilon))$, and the maximal influence of $f$ is $I_{max}(f)=Theta(epsilon min(1,a' sqrt{log(1/epsilon)}))$. \u0000These results, which determine the exact asymptotic order of $W^1(f)$ and $I_{max}(f)$, provide sharp generalizations of theorems proved by Matulef, O'Donnell, Rubinfeld, and Servedio, and settle a conjecture posed by Kalai, Keller and Mossel. \u0000In addition, we present a refinement of the definition of noise sensitivity which takes into consideration the bias of the function, and show that (like in the unbiased case) halfspaces are noise resistant, and, in the other direction, any noise resistant function is well correlated with a halfspace. \u0000Our main tools are 'local' forms of the classical Chernoff inequality, like the following one proved by Devroye and Lugosi (2008): \u0000Let ${ x_i }$ be independent random variables uniformly distributed in ${-1,1}$, and let $a_iinmathbb{R}_+$ be such that $sum_i a_{i}^{2}=1$. \u0000If for some $tgeq 0$ we have $Pr[sum_{i} a_i x_i > t]=epsilon$, then $Pr[sum_{i} a_i x_i>t+delta]leq frac{epsilon}{2}$ holds for $deltaleq c/sqrt{log(1/epsilon)}$, where $c$ is a universal constant.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47104343","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 a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-gamma)|S||A|$ with a real $gammain(0,1]$, then $|A| ge 4^{(1-1/d)gamma |S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$, our bound translates into a sharp estimate for the additive dimension of the popular difference set. We also prove, as an auxiliary result, the following estimate of possible independent interest: if $A subset mathbb Z^n$ is a finite, non-empty downset then, denoting by $w(a)$ the number of non-zero components of the vector $ain A$, we have [frac1{|A|} sum_{ain A} w(a) le frac12, log_2 |A|.]
{"title":"On Isoperimetric Stability","authors":"V. Lev","doi":"10.19086/DA.3699","DOIUrl":"https://doi.org/10.19086/DA.3699","url":null,"abstract":"We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-gamma)|S||A|$ with a real $gammain(0,1]$, then $|A| ge 4^{(1-1/d)gamma |S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. \u0000As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$, our bound translates into a sharp estimate for the additive dimension of the popular difference set. \u0000We also prove, as an auxiliary result, the following estimate of possible independent interest: if $A subset mathbb Z^n$ is a finite, non-empty downset then, denoting by $w(a)$ the number of non-zero components of the vector $ain A$, we have \u0000[frac1{|A|} sum_{ain A} w(a) le frac12, log_2 |A|.]","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49377718","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 semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1, ldots, x_n)$ be a degree $d$ homogeneous form with integer coefficients. We provide sufficient conditions, similar to that of the seminal work of B. J. Birch, for which the equation $F (x_1, ldots, x_n) = 0$ has infinitely many solutions whose coordinates are all semiprimes. Previously it was known due to 'A. Magyar and T. Titichetrakun that under the same hypotheses there exist infinite number of integer solutions to the equation whose coordinates have at most $384 n^{3/2} d (d+1)$ prime factors. Our main result reduces this bound on the number of prime factors from $384 n^{3/2} d (d+1)$ to $2$.
{"title":"Diophantine equations in semiprimes","authors":"S. Yamagishi","doi":"10.19086/da.11075","DOIUrl":"https://doi.org/10.19086/da.11075","url":null,"abstract":"A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let $F(x_1, ldots, x_n)$ be a degree $d$ homogeneous form with integer coefficients. We provide sufficient conditions, similar to that of the seminal work of B. J. Birch, for which the equation $F (x_1, ldots, x_n) = 0$ has infinitely many solutions whose coordinates are all semiprimes. Previously it was known due to 'A. Magyar and T. Titichetrakun that under the same hypotheses there exist infinite number of integer solutions to the equation whose coordinates have at most $384 n^{3/2} d (d+1)$ prime factors. Our main result reduces this bound on the number of prime factors from $384 n^{3/2} d (d+1)$ to $2$.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46459426","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 consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope.
{"title":"Floating and Illumination Bodies for Polytopes: Duality Results.","authors":"Olaf Mordhorst, E. Werner","doi":"10.19086/DA.8973","DOIUrl":"https://doi.org/10.19086/DA.8973","url":null,"abstract":"We consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44493377","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}
Magnitude is an invariant of metric spaces with origins in enriched category theory. Using potential theoretic methods, Barcelo and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schroder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows.
{"title":"The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials","authors":"S. Willerton","doi":"10.19086/da.12649","DOIUrl":"https://doi.org/10.19086/da.12649","url":null,"abstract":"Magnitude is an invariant of metric spaces with origins in enriched category theory. Using potential theoretic methods, Barcelo and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schroder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43284275","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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