In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of integers is represented as such a sum, where the parameters are restricted to have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$ is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is a natural linear function in $n$.
{"title":"Universal sums of generalized polygonal numbers of almost prime \"length\"","authors":"Soumyarup Banerjee, Ben Kane, Daejun Kim","doi":"arxiv-2409.07895","DOIUrl":"https://doi.org/arxiv-2409.07895","url":null,"abstract":"In this paper, we consider sums of three generalized $m$-gonal numbers whose\u0000parameters are restricted to integers with a bounded number of prime divisors.\u0000With some restrictions on $m$ modulo $30$, we show that a density one set of\u0000integers is represented as such a sum, where the parameters are restricted to\u0000have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$\u0000is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is\u0000a natural linear function in $n$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(frac {j+k}p)pm(frac{j-k}p)$, where $(frac{cdot}p)$ denotes the Legendre symbol. When $pequiv1pmod4$, we determine the characteristic polynomials of the matrices $$left[left(frac{j+k}pright)+left(frac{j-k}pright)right]_{1le j,kle n} text{and} left[left(frac{j+k}pright)-left(frac{j-k}pright)right]_{1le j,kle n},$$ and also prove that begin{align*} & left|x+left(frac{j+k}pright)+left(frac{j-k}pright)+left(frac jpright)y+left(frac kpright)z+left(frac{jk}pright)wright|_{1le j,kle n} =& (-p)^{(p-5)/4}left(left(frac{p-1}2right)^2wx-left(frac{p-1}2y-1right)left(frac{p-1}2z-1right)right), end{align*} which was previously conjectured by the second author.
{"title":"On determinants involving $(frac{j+k}p)pm(frac{j-k}p)$","authors":"Deyi Chen, Zhi-Wei Sun","doi":"arxiv-2409.08213","DOIUrl":"https://doi.org/arxiv-2409.08213","url":null,"abstract":"Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants\u0000involving $(frac {j+k}p)pm(frac{j-k}p)$, where $(frac{cdot}p)$ denotes the\u0000Legendre symbol. When $pequiv1pmod4$, we determine the characteristic\u0000polynomials of the matrices\u0000$$left[left(frac{j+k}pright)+left(frac{j-k}pright)right]_{1le j,kle\u0000n} text{and} \u0000left[left(frac{j+k}pright)-left(frac{j-k}pright)right]_{1le j,kle\u0000n},$$ and also prove that begin{align*} &\u0000left|x+left(frac{j+k}pright)+left(frac{j-k}pright)+left(frac\u0000jpright)y+left(frac kpright)z+left(frac{jk}pright)wright|_{1le j,kle\u0000n} =&\u0000(-p)^{(p-5)/4}left(left(frac{p-1}2right)^2wx-left(frac{p-1}2y-1right)left(frac{p-1}2z-1right)right),\u0000end{align*} which was previously conjectured by the second author.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"298 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove an effective version of the inverse theorem for the Gowers $U^3$-norm for functions supported on high-rank quadratic level sets in finite vector spaces. For configurations controlled by the $U^3$-norm (complexity-two configurations), this enables one to run a density increment argument with respect to quadratic level sets, which are analogues of Bohr sets in the context of quadratic Fourier analysis on finite vector spaces. We demonstrate such an argument by deriving an exponential bound on the Ramsey number of three-term progressions which are the same colour as their common difference (``Brauer quadruples''), a result we have been unable to establish by other means. Our methods also yield polylogarithmic bounds on the density of sets lacking translation-invariant configurations of complexity two. Such bounds for four-term progressions were obtained by Green and Tao using a simpler weak-regularity argument. In an appendix, we give an example of how to generalise Green and Tao's argument to other translation-invariant configurations of complexity two. However, this crucially relies on an estimate coming from the Croot-Lev-Pach polynomial method, which may not be applicable to all systems of complexity two. Hence running a density increment with respect to quadratic level sets may still prove useful for such problems. It may also serve as a model for running density increments on more general nil-Bohr sets, with a view to effectivising other Szemer'edi-type theorems.
{"title":"An inverse theorem for the Gowers $U^3$-norm relative to quadratic level sets","authors":"Sean Prendiville","doi":"arxiv-2409.07962","DOIUrl":"https://doi.org/arxiv-2409.07962","url":null,"abstract":"We prove an effective version of the inverse theorem for the Gowers\u0000$U^3$-norm for functions supported on high-rank quadratic level sets in finite\u0000vector spaces. For configurations controlled by the $U^3$-norm (complexity-two\u0000configurations), this enables one to run a density increment argument with\u0000respect to quadratic level sets, which are analogues of Bohr sets in the\u0000context of quadratic Fourier analysis on finite vector spaces. We demonstrate\u0000such an argument by deriving an exponential bound on the Ramsey number of\u0000three-term progressions which are the same colour as their common difference\u0000(``Brauer quadruples''), a result we have been unable to establish by other\u0000means. Our methods also yield polylogarithmic bounds on the density of sets lacking\u0000translation-invariant configurations of complexity two. Such bounds for\u0000four-term progressions were obtained by Green and Tao using a simpler\u0000weak-regularity argument. In an appendix, we give an example of how to\u0000generalise Green and Tao's argument to other translation-invariant\u0000configurations of complexity two. However, this crucially relies on an estimate\u0000coming from the Croot-Lev-Pach polynomial method, which may not be applicable\u0000to all systems of complexity two. Hence running a density increment with\u0000respect to quadratic level sets may still prove useful for such problems. It\u0000may also serve as a model for running density increments on more general\u0000nil-Bohr sets, with a view to effectivising other Szemer'edi-type theorems.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In his paper on the Mordell-Lang conjecture, Hrushovski employed techniques from model theory to prove the function field version of the conjecture. In doing so he was able to answer a related question of Voloch, which we refer to henceforth as Hrushovski's theorem. In this paper we shall give an alternative proof of said theorem in the characteristic $p$ setting, but using purely algebro-geometric ideas.
{"title":"An Algebraic Proof of Hrushovski's Theorem","authors":"Thomas Wisson","doi":"arxiv-2409.08370","DOIUrl":"https://doi.org/arxiv-2409.08370","url":null,"abstract":"In his paper on the Mordell-Lang conjecture, Hrushovski employed techniques\u0000from model theory to prove the function field version of the conjecture. In\u0000doing so he was able to answer a related question of Voloch, which we refer to\u0000henceforth as Hrushovski's theorem. In this paper we shall give an alternative\u0000proof of said theorem in the characteristic $p$ setting, but using purely\u0000algebro-geometric ideas.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we provide evidence for a certain twisted version of the parity conjecture for Jacobians, introduced in prior work of V. Dokchitser, Green, Konstantinou and the author. To do this, we use arithmetic duality theorems for abelian varieties to study the determinant of certain endomorphisms acting on p-infinity Selmer groups.
在本注释中，我们为雅各布数的奇偶性猜想的某个扭曲版本提供了证据，该猜想是在 V. Dokchitser、Green、Konstantinou 和作者的先前工作中引入的。为此，我们利用阿贝尔变项的算术对偶定理来研究作用于 p 无穷塞尔默群的某些内定形的行列式。
{"title":"A note on local formulae for the parity of Selmer ranks","authors":"Adam Morgan","doi":"arxiv-2409.08034","DOIUrl":"https://doi.org/arxiv-2409.08034","url":null,"abstract":"In this note, we provide evidence for a certain twisted version of the parity\u0000conjecture for Jacobians, introduced in prior work of V. Dokchitser, Green,\u0000Konstantinou and the author. To do this, we use arithmetic duality theorems for\u0000abelian varieties to study the determinant of certain endomorphisms acting on\u0000p-infinity Selmer groups.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let an odd integer (mathcal{X}) be expressed as $left{sumlimits_{M > m}b_M2^Mright}+2^m-1,$ where $b_Min{0,1}$ and $2^m-1$ is referred to as the Governor. In Collatz-type functions, a high index Governor is eventually reduced to $2^1-1$. For the $3mathcal{Z}+1$ sequence, the Governor occurring in the Trivial cycle is $2^1-1$, while for the $5mathcal{Z}+1$ sequence, the Trivial Governors are $2^2-1$ and $2^1-1$. Therefore, in these specific sequences, the Collatz function reduces the Governor $2^m - 1$ to the Trivial Governor $2^{mathcal{T}} - 1$. Once this Trivial Governor is reached, it can evolve to a higher index Governor through interactions with other terms. This feature allows $mathcal{X}$ to reappear in a Collatz-type sequence, since $2^m - 1 = 2^{m - 1} + cdots + 2^{mathcal{T} + 1} + 2^{mathcal{T}}+(2^{mathcal{T}}-1).$ Thus, if $mathcal{X}$ reappears, at least one odd ancestor of $left{sumlimits_{M > m}b_M2^Mright}+2^{m-1}+cdots+2^{mathcal{T}+1}+2^{mathcal{T}}+(2^{mathcal{T}}-1)$ must have the Governor $2^m-1$. Ancestor mapping shows that all odd ancestors of $mathcal{X}$ have the Trivial Governor for the respective Collatz sequence. This implies that odd integers that repeat in the $3mathcal{Z} + 1$ sequence have the Governor $2^1 - 1$, while those forming a repeating cycle in the $5mathcal{Z} + 1$ sequence have either $2^2 - 1$ or $2^1 - 1$ as the Governor. Successor mapping for the $3mathcal{Z} + 1$ sequence further indicates that there are no auxiliary cycles, as the Trivial Governor is always transformed into a different index Governor. Similarly, successor mapping for the $5mathcal{Z} + 1$ sequence reveals that the smallest odd integers forming an auxiliary cycle are smaller than $2^5$. Finally, attempts to identify integers that diverge for the $3mathcal{Z} + 1$ sequence suggest that no such integers exist.
{"title":"General Dynamics and Generation Mapping for Collatz-type Sequences","authors":"Gaurav Goyal","doi":"arxiv-2409.07929","DOIUrl":"https://doi.org/arxiv-2409.07929","url":null,"abstract":"Let an odd integer (mathcal{X}) be expressed as $left{sumlimits_{M >\u0000m}b_M2^Mright}+2^m-1,$ where $b_Min{0,1}$ and $2^m-1$ is referred to as\u0000the Governor. In Collatz-type functions, a high index Governor is eventually\u0000reduced to $2^1-1$. For the $3mathcal{Z}+1$ sequence, the Governor occurring\u0000in the Trivial cycle is $2^1-1$, while for the $5mathcal{Z}+1$ sequence, the\u0000Trivial Governors are $2^2-1$ and $2^1-1$. Therefore, in these specific\u0000sequences, the Collatz function reduces the Governor $2^m - 1$ to the Trivial\u0000Governor $2^{mathcal{T}} - 1$. Once this Trivial Governor is reached, it can\u0000evolve to a higher index Governor through interactions with other terms. This\u0000feature allows $mathcal{X}$ to reappear in a Collatz-type sequence, since $2^m\u0000- 1 = 2^{m - 1} + cdots + 2^{mathcal{T} + 1} +\u00002^{mathcal{T}}+(2^{mathcal{T}}-1).$ Thus, if $mathcal{X}$ reappears, at\u0000least one odd ancestor of $left{sumlimits_{M >\u0000m}b_M2^Mright}+2^{m-1}+cdots+2^{mathcal{T}+1}+2^{mathcal{T}}+(2^{mathcal{T}}-1)$\u0000must have the Governor $2^m-1$. Ancestor mapping shows that all odd ancestors\u0000of $mathcal{X}$ have the Trivial Governor for the respective Collatz sequence.\u0000This implies that odd integers that repeat in the $3mathcal{Z} + 1$ sequence\u0000have the Governor $2^1 - 1$, while those forming a repeating cycle in the\u0000$5mathcal{Z} + 1$ sequence have either $2^2 - 1$ or $2^1 - 1$ as the Governor.\u0000Successor mapping for the $3mathcal{Z} + 1$ sequence further indicates that\u0000there are no auxiliary cycles, as the Trivial Governor is always transformed\u0000into a different index Governor. Similarly, successor mapping for the\u0000$5mathcal{Z} + 1$ sequence reveals that the smallest odd integers forming an\u0000auxiliary cycle are smaller than $2^5$. Finally, attempts to identify integers\u0000that diverge for the $3mathcal{Z} + 1$ sequence suggest that no such integers\u0000exist.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We demonstrate equidistribution of the lattice shape of cubic fields when ordered by discriminant, giving an estimate in the Eisenstein series spectrum with a lower order main term. The analysis gives a separate discussion of the contributions of reducible and irreducible binary cubic forms, following a method of Shintani. Our work answers a question posed at the American Institute of Math by giving a precise geometric and spectral description of an evident barrier to equidistribution in the lattice shape.
{"title":"Lower order terms in the shape of cubic fields","authors":"Robert Hough, Eun Hye Lee","doi":"arxiv-2409.08417","DOIUrl":"https://doi.org/arxiv-2409.08417","url":null,"abstract":"We demonstrate equidistribution of the lattice shape of cubic fields when\u0000ordered by discriminant, giving an estimate in the Eisenstein series spectrum\u0000with a lower order main term. The analysis gives a separate discussion of the\u0000contributions of reducible and irreducible binary cubic forms, following a\u0000method of Shintani. Our work answers a question posed at the American Institute\u0000of Math by giving a precise geometric and spectral description of an evident\u0000barrier to equidistribution in the lattice shape.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ondřej ChwiedziukCharles University, Matěj DoležálekCharles University, Simona HlavinkováCharles University, Emma PěchoučkováCharles University, Zdeněk PezlarCharles University, Om PrakashCharles University, Anna RůžičkováCharles University, Mikuláš ZindulkaCharles University
We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is universal. We also construct an example illustrating that the positive-definiteness condition is necessary.
{"title":"No proper generalized quadratic forms are universal over quadratic fields","authors":"Ondřej ChwiedziukCharles University, Matěj DoležálekCharles University, Simona HlavinkováCharles University, Emma PěchoučkováCharles University, Zdeněk PezlarCharles University, Om PrakashCharles University, Anna RůžičkováCharles University, Mikuláš ZindulkaCharles University","doi":"arxiv-2409.07941","DOIUrl":"https://doi.org/arxiv-2409.07941","url":null,"abstract":"We consider generalized quadratic forms over real quadratic number fields and\u0000prove, under a natural positive-definiteness condition, that a generalized\u0000quadratic form can only be universal if it contains a quadratic subform that is\u0000universal. We also construct an example illustrating that the\u0000positive-definiteness condition is necessary.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider sums of values of degenerate falling factorials and give a probabilistic proof of a recurrence relation for them. This may be viewed as a degenerate version of the recent probabilistic proofs on sums of powers of integers.
{"title":"A probabilistic proof of a recurrence relation for sums of values of degenerate falling factorials","authors":"Taekyun Kim, Dae san Kim","doi":"arxiv-2409.07742","DOIUrl":"https://doi.org/arxiv-2409.07742","url":null,"abstract":"In this paper, we consider sums of values of degenerate falling factorials\u0000and give a probabilistic proof of a recurrence relation for them. This may be\u0000viewed as a degenerate version of the recent probabilistic proofs on sums of\u0000powers of integers.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show the following result: If $X_0$ is an affine surface over a field $K$ and $f, g$ are two loxodromic automorphisms with an orbit meeting infinitely many times, then $f$ and $g$ must share a common iterate. The proof uses the preliminary work of the author in [Abb23] on the dynamics of endomorphisms of affine surfaces and arguments from arithmetic dynamics. We then show a dynamical Mordell-Lang type result for surfaces in $X_0 times X_0$.
{"title":"Intersection of orbits of loxodromic automorphisms of affine surfaces","authors":"Marc Abboud","doi":"arxiv-2409.07826","DOIUrl":"https://doi.org/arxiv-2409.07826","url":null,"abstract":"We show the following result: If $X_0$ is an affine surface over a field $K$\u0000and $f, g$ are two loxodromic automorphisms with an orbit meeting infinitely\u0000many times, then $f$ and $g$ must share a common iterate. The proof uses the\u0000preliminary work of the author in [Abb23] on the dynamics of endomorphisms of\u0000affine surfaces and arguments from arithmetic dynamics. We then show a\u0000dynamical Mordell-Lang type result for surfaces in $X_0 times X_0$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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