Abstract We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.
{"title":"Embedding codimension of the space of arcs","authors":"C. Chiu, Tommaso de Fernex, Roi Docampo","doi":"10.1017/fmp.2021.19","DOIUrl":"https://doi.org/10.1017/fmp.2021.19","url":null,"abstract":"Abstract We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48911737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $lambda _g$-insertion is related to Gromov-Witten theory of the total space of ${mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={mathbb P}^2$, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${mathbb P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${mathbb P}^2$ in the Nekrasov-Shatashvili limit.
摘要证明了van Garrel-Graber-Ruddat的格$0$局部相对对应的一个高格版本:对于$(X,D)$ a对，其中X是光滑投影变量，D是nef光滑因子，$(X,D)$与$lambda _g$插入的最大接触Gromov-Witten理论与${mathcal O}_X(-D)$的总空间的Gromov-Witten理论和D的局部Gromov-Witten理论有关。对于S a del Pezzo曲面或有理椭圆曲面，E是光滑反正则因子，专门讨论$(X,D)=(S,E)$。我们证明了$(S,E)$的极大接触Gromov-Witten理论是由Calabi-Yau 3-fold ${ mathbb P}^2$的平稳Gromov-Witten理论和$({mathbb P}^2,E)$的极大接触Gromov-Witten不变量的高格生成级数是准模的，满足全纯异常方程。该证明结合了准模性结果和先前已知的局部${mathbb P}^2$和椭圆曲线的全纯异常方程。进一步，利用$({mathbb P}^2,E)$的最大接触Gromov-Witten不变量与${mathbb P}^2$上半稳定一维束模空间的Betti数之间的联系，证明了物理文献中预测的局部${mathbb P}^2$的精化拓扑弦自由能在Nekrasov-Shatashvili极限下的准模性和全纯异常方程。
{"title":"Holomorphic anomaly equation for $({mathbb P}^2,E)$ and the Nekrasov-Shatashvili limit of local ${mathbb P}^2$","authors":"Pierrick Bousseau, H. Fan, Shuai Guo, Longting Wu","doi":"10.1017/fmp.2021.3","DOIUrl":"https://doi.org/10.1017/fmp.2021.3","url":null,"abstract":"Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $lambda _g$-insertion is related to Gromov-Witten theory of the total space of ${mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={mathbb P}^2$, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${mathbb P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${mathbb P}^2$ in the Nekrasov-Shatashvili limit.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2021.3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43779572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group ${mathbf G}$ depends on two parameters: the $sigma $ -conjugacy class $[b]$ and the element w in the Iwahori–Weyl group $tilde {W}$ of ${mathbf G}$ . In this paper, for any given $sigma $ -conjugacy class $[b]$ , we determine the nonemptiness pattern and the dimension formula of $X_w(b)$ for most $w in tilde {W}$ .
{"title":"Cordial elements and dimensions of affine Deligne–Lusztig varieties","authors":"Xuhua He","doi":"10.1017/fmp.2021.10","DOIUrl":"https://doi.org/10.1017/fmp.2021.10","url":null,"abstract":"Abstract The affine Deligne–Lusztig variety \u0000$X_w(b)$\u0000 in the affine flag variety of a reductive group \u0000${mathbf G}$\u0000 depends on two parameters: the \u0000$sigma $\u0000 -conjugacy class \u0000$[b]$\u0000 and the element w in the Iwahori–Weyl group \u0000$tilde {W}$\u0000 of \u0000${mathbf G}$\u0000 . In this paper, for any given \u0000$sigma $\u0000 -conjugacy class \u0000$[b]$\u0000 , we determine the nonemptiness pattern and the dimension formula of \u0000$X_w(b)$\u0000 for most \u0000$w in tilde {W}$\u0000 .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44654971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $T_{mathrm {kin}} gg 1$ and in a limiting regime where the size L of the domain goes to infinity and the strength $alpha $ of the nonlinearity goes to $0$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $T_{mathrm {kin}}=Oleft (alpha ^{-2}right )$ and $alpha $ is related to the conserved mass $lambda $ of the solution via $alpha =lambda ^2 L^{-d}$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $(alpha , L)$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $alpha $ approaches $0$ like $L^{-varepsilon +}$ or like $L^{-1-frac {varepsilon }{2}+}$ (for arbitrary small $varepsilon $ ), we exhibit the wave kinetic equation up to time scales $O(T_{mathrm {kin}}L^{-varepsilon })$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $T_*ll T_{mathrm {kin}}$ and identify specific interactions that become very large for times beyond $T_*$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $T_*$ toward $T_{mathrm {kin}}$ for such scaling laws seems to require new methods and ideas.
{"title":"On the derivation of the wave kinetic equation for NLS","authors":"Yu Deng, Z. Hani","doi":"10.1017/fmp.2021.6","DOIUrl":"https://doi.org/10.1017/fmp.2021.6","url":null,"abstract":"Abstract A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale \u0000$T_{mathrm {kin}} gg 1$\u0000 and in a limiting regime where the size L of the domain goes to infinity and the strength \u0000$alpha $\u0000 of the nonlinearity goes to \u0000$0$\u0000 (weak nonlinearity). For the cubic nonlinear Schrödinger equation, \u0000$T_{mathrm {kin}}=Oleft (alpha ^{-2}right )$\u0000 and \u0000$alpha $\u0000 is related to the conserved mass \u0000$lambda $\u0000 of the solution via \u0000$alpha =lambda ^2 L^{-d}$\u0000 . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the \u0000$(alpha , L)$\u0000 limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when \u0000$alpha $\u0000 approaches \u0000$0$\u0000 like \u0000$L^{-varepsilon +}$\u0000 or like \u0000$L^{-1-frac {varepsilon }{2}+}$\u0000 (for arbitrary small \u0000$varepsilon $\u0000 ), we exhibit the wave kinetic equation up to time scales \u0000$O(T_{mathrm {kin}}L^{-varepsilon })$\u0000 , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales \u0000$T_*ll T_{mathrm {kin}}$\u0000 and identify specific interactions that become very large for times beyond \u0000$T_*$\u0000 . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond \u0000$T_*$\u0000 toward \u0000$T_{mathrm {kin}}$\u0000 for such scaling laws seems to require new methods and ideas.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2021.6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49053477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dennis Eriksson, Gerard Freixas i Montplet, Christophe Mourougane
Abstract The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé and our previous results on the BCOV invariant, we establish this conjecture for Calabi–Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang–Lu–Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla–Selberg type theorem expressing it in terms of special $Gamma $ -values for certain Calabi–Yau manifolds with complex multiplication.
{"title":"On genus one mirror symmetry in higher dimensions and the BCOV conjectures","authors":"Dennis Eriksson, Gerard Freixas i Montplet, Christophe Mourougane","doi":"10.1017/fmp.2022.13","DOIUrl":"https://doi.org/10.1017/fmp.2022.13","url":null,"abstract":"Abstract The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé and our previous results on the BCOV invariant, we establish this conjecture for Calabi–Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang–Lu–Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla–Selberg type theorem expressing it in terms of special \u0000$Gamma $\u0000 -values for certain Calabi–Yau manifolds with complex multiplication.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42518767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi
Abstract We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.
{"title":"Hensel minimality I","authors":"R. Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi","doi":"10.1017/fmp.2022.6","DOIUrl":"https://doi.org/10.1017/fmp.2022.6","url":null,"abstract":"Abstract We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41458600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{overline {K}}$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.
{"title":"Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields","authors":"A. Shankar, A. Shankar, Yunqing Tang, Salim Tayou","doi":"10.1017/fmp.2022.14","DOIUrl":"https://doi.org/10.1017/fmp.2022.14","url":null,"abstract":"Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either \u0000$X_{overline {K}}$\u0000 has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44277080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Define the Collatz map ${operatorname {Col}} colon mathbb {N}+1 to mathbb {N}+1$ on the positive integers $mathbb {N}+1 = {1,2,3,dots }$ by setting ${operatorname {Col}}(N)$ equal to $3N+1$ when N is odd and $N/2$ when N is even, and let ${operatorname {Col}}_{min }(N) := inf _{n in mathbb {N}} {operatorname {Col}}^n(N)$ denote the minimal element of the Collatz orbit $N, {operatorname {Col}}(N), {operatorname {Col}}^2(N), dots $ . The infamous Collatz conjecture asserts that ${operatorname {Col}}_{min }(N)=1$ for all $N in mathbb {N}+1$ . Previously, it was shown by Korec that for any $theta> frac {log 3}{log 4} approx 0.7924$ , one has ${operatorname {Col}}_{min }(N) leq N^theta $ for almost all $N in mathbb {N}+1$ (in the sense of natural density). In this paper, we show that for any function $f colon mathbb {N}+1 to mathbb {R}$ with $lim _{N to infty } f(N)=+infty $ , one has ${operatorname {Col}}_{min }(N) leq f(N)$ for almost all $N in mathbb {N}+1$ (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a $3$ -adic cyclic group $mathbb {Z}/3^nmathbb {Z}$ at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.
{"title":"Almost all orbits of the Collatz map attain almost bounded values","authors":"T. Tao","doi":"10.1017/fmp.2022.8","DOIUrl":"https://doi.org/10.1017/fmp.2022.8","url":null,"abstract":"Abstract Define the Collatz map \u0000${operatorname {Col}} colon mathbb {N}+1 to mathbb {N}+1$\u0000 on the positive integers \u0000$mathbb {N}+1 = {1,2,3,dots }$\u0000 by setting \u0000${operatorname {Col}}(N)$\u0000 equal to \u0000$3N+1$\u0000 when N is odd and \u0000$N/2$\u0000 when N is even, and let \u0000${operatorname {Col}}_{min }(N) := inf _{n in mathbb {N}} {operatorname {Col}}^n(N)$\u0000 denote the minimal element of the Collatz orbit \u0000$N, {operatorname {Col}}(N), {operatorname {Col}}^2(N), dots $\u0000 . The infamous Collatz conjecture asserts that \u0000${operatorname {Col}}_{min }(N)=1$\u0000 for all \u0000$N in mathbb {N}+1$\u0000 . Previously, it was shown by Korec that for any \u0000$theta> frac {log 3}{log 4} approx 0.7924$\u0000 , one has \u0000${operatorname {Col}}_{min }(N) leq N^theta $\u0000 for almost all \u0000$N in mathbb {N}+1$\u0000 (in the sense of natural density). In this paper, we show that for any function \u0000$f colon mathbb {N}+1 to mathbb {R}$\u0000 with \u0000$lim _{N to infty } f(N)=+infty $\u0000 , one has \u0000${operatorname {Col}}_{min }(N) leq f(N)$\u0000 for almost all \u0000$N in mathbb {N}+1$\u0000 (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a \u0000$3$\u0000 -adic cyclic group \u0000$mathbb {Z}/3^nmathbb {Z}$\u0000 at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44030374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let $P_1,dots ,P_min mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of ${1,dots ,N}$ with no nontrivial progressions of the form $x,x+P_1(y),dots ,x+P_m(y)$ has size $|A|ll N/(log log {N})^{c_{P_1,dots ,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.
{"title":"Bounds for sets with no polynomial progressions","authors":"Sarah Peluse","doi":"10.1017/fmp.2020.11","DOIUrl":"https://doi.org/10.1017/fmp.2020.11","url":null,"abstract":"Abstract Let $P_1,dots ,P_min mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of ${1,dots ,N}$ with no nontrivial progressions of the form $x,x+P_1(y),dots ,x+P_m(y)$ has size $|A|ll N/(log log {N})^{c_{P_1,dots ,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.11","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45656312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.
{"title":"Smoothing toroidal crossing spaces","authors":"Simon Felten, Matej Filip, Helge Ruddat","doi":"10.1017/fmp.2021.8","DOIUrl":"https://doi.org/10.1017/fmp.2021.8","url":null,"abstract":"Abstract We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44242862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：