Pub Date : 2024-11-20DOI: 10.1016/j.jnt.2024.10.008
Sebastian Monnet
Given a 2-adic field K, we give a formula for the number of totally ramified quartic field extensions with a given discriminant valuation and Galois closure group. We use these formulae to prove refinements of Serre's mass formula, which will have applications to the arithmetic statistics of number fields.
{"title":"Counting wild quartics with prescribed discriminant and Galois closure group","authors":"Sebastian Monnet","doi":"10.1016/j.jnt.2024.10.008","DOIUrl":"10.1016/j.jnt.2024.10.008","url":null,"abstract":"<div><div>Given a 2-adic field <em>K</em>, we give a formula for the number of totally ramified quartic field extensions <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> with a given discriminant valuation and Galois closure group. We use these formulae to prove refinements of Serre's mass formula, which will have applications to the arithmetic statistics of number fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 157-202"},"PeriodicalIF":0.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.jnt.2024.09.015
Ernst-Ulrich Gekeler
We study expansions of Drinfeld modular forms of rank along the boundary of moduli varieties. Product formulas for the discriminant forms are developed, which are analogous with Jacobi's formula for the classical elliptic discriminant. The vanishing orders are described through values at of partial zeta functions of the underlying Drinfeld coefficient ring A. We show linear independence properties for Eisenstein series, which allow to split spaces of modular forms into the subspaces of cusp forms and of Eisenstein series, and give various characterizations of the boundary condition for modular forms.
{"title":"On Drinfeld modular forms of higher rank VII: Expansions at the boundary","authors":"Ernst-Ulrich Gekeler","doi":"10.1016/j.jnt.2024.09.015","DOIUrl":"10.1016/j.jnt.2024.09.015","url":null,"abstract":"<div><div>We study expansions of Drinfeld modular forms of rank <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> along the boundary of moduli varieties. Product formulas for the discriminant forms <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are developed, which are analogous with Jacobi's formula for the classical elliptic discriminant. The vanishing orders are described through values at <span><math><mi>s</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>r</mi></math></span> of partial zeta functions of the underlying Drinfeld coefficient ring <em>A</em>. We show linear independence properties for Eisenstein series, which allow to split spaces of modular forms into the subspaces of cusp forms and of Eisenstein series, and give various characterizations of the boundary condition for modular forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 260-340"},"PeriodicalIF":0.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.jnt.2024.10.001
Chiara Bellotti
We will provide an explicit log-free zero-density estimate for of the form . In particular, this estimate becomes the sharpest known explicit zero-density estimate uniformly for , with and .
{"title":"An explicit log-free zero density estimate for the Riemann zeta-function","authors":"Chiara Bellotti","doi":"10.1016/j.jnt.2024.10.001","DOIUrl":"10.1016/j.jnt.2024.10.001","url":null,"abstract":"<div><div>We will provide an explicit log-free zero-density estimate for <span><math><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> of the form <span><math><mi>N</mi><mo>(</mo><mi>σ</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>≤</mo><mi>A</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>B</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>σ</mi><mo>)</mo></mrow></msup></math></span>. In particular, this estimate becomes the sharpest known explicit zero-density estimate uniformly for <span><math><mi>σ</mi><mo>∈</mo><mo>[</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, with <span><math><mn>0.985</mn><mo>≤</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≤</mo><mn>0.9927</mn></math></span> and <span><math><mn>3</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup><mo><</mo><mi>T</mi><mo>≤</mo><mi>exp</mi><mo></mo><mo>(</mo><mn>6.7</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 37-77"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.jnt.2024.10.003
Bing He, Shuming Liu
In this paper, we will study a conjecture of Merca on theta series, which gives a refinement of a conjecture of Andrews and Merca on truncated pentagonal number series. We first show refinements of two special cases of Merca's conjecture and then establish several nonnegativity results on theta series. As applications, we establish positivity results involving two celebrated partition statistics.
{"title":"A conjecture of Merca on nonnegativity of theta series","authors":"Bing He, Shuming Liu","doi":"10.1016/j.jnt.2024.10.003","DOIUrl":"10.1016/j.jnt.2024.10.003","url":null,"abstract":"<div><div>In this paper, we will study a conjecture of Merca on theta series, which gives a refinement of a conjecture of Andrews and Merca on truncated pentagonal number series. We first show refinements of two special cases of Merca's conjecture and then establish several nonnegativity results on theta series. As applications, we establish positivity results involving two celebrated partition statistics.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 17-36"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744195","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}
Pub Date : 2024-11-19DOI: 10.1016/j.jnt.2024.10.002
Francesco Cellarosi , Tariq Osman
We provide explicit families of pairs such that for sufficiently regular f, there is a constant C for which the theta sum bound holds for every and every . Central to the proof is realising that, for fixed N, the theta sum normalised by agrees with an automorphic function evaluated along a special curve known as a horocycle lift. The lift depends on the pair , and so the bound follows from showing that there are pairs such that remains bounded along the entire horocycle lift.
{"title":"Bounds for smooth theta sums with rational parameters","authors":"Francesco Cellarosi , Tariq Osman","doi":"10.1016/j.jnt.2024.10.002","DOIUrl":"10.1016/j.jnt.2024.10.002","url":null,"abstract":"<div><div>We provide explicit families of pairs <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> such that for sufficiently regular <em>f</em>, there is a constant <em>C</em> for which the theta sum bound<span><span><span><math><mrow><mo>|</mo><munder><mo>∑</mo><mrow><mtext>n</mtext><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mi>f</mi><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mrow><mi>n</mi></mrow><mo>)</mo></mrow><mi>exp</mi><mo></mo><mo>{</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo>(</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mo>‖</mo><mrow><mi>n</mi></mrow><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi>β</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mi>x</mi><mo>+</mo><mrow><mi>α</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mo>}</mo><mo>|</mo></mrow><mspace></mspace><mo>≤</mo><mi>C</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> holds for every <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> and every <span><math><mi>N</mi><mo>∈</mo><mi>N</mi></math></span>. Central to the proof is realising that, for fixed <em>N</em>, the theta sum normalised by <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> agrees with an automorphic function <span><math><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> evaluated along a special curve known as a horocycle lift. The lift depends on the pair <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo></math></span>, and so the bound follows from showing that there are pairs such that <span><math><mo>|</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>|</mo></math></span> remains bounded along the entire horocycle lift.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 397-426"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744088","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}
Pub Date : 2024-11-06DOI: 10.1016/j.jnt.2024.09.002
Kavita Dhanda, Alan Haynes
Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points with and . In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of α whose accumulation points are all of . In the second part we focus primarily on the case when the coordinates of α together with 1 form a basis for an algebraic number field K. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when , this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in K) of a single ellipse, or of a pair of hyperbolas, depending on whether or not K has a non-trivial embedding into .
以二阶近似理论的经典方面为基础，我们考虑了具有 α∈Rd 和 q∈Z 的点 qα 的归一化整数向量平移的所有堆积点的集合。在论文的第一部分，我们推导了关于积点都是 Rd 的 α 集合的度量论和豪斯多夫维度结果。在第二部分中，我们主要关注当 α 的坐标与 1 一起构成代数数域 K 的基础时的情况。我们在此证明，在正确的归一化条件下，累积点集合显示出有序的几何结构，它反映了基础数域的代数特性。例如，当 d=2 时，这个积点集合可以被描述为一个椭圆或一对双曲线的扩张（通过 K 中一个阶元素的规范）的可数联合，这取决于 K 是否有一个非三维嵌入到 C 中。
{"title":"Accumulation points of normalized approximations","authors":"Kavita Dhanda, Alan Haynes","doi":"10.1016/j.jnt.2024.09.002","DOIUrl":"10.1016/j.jnt.2024.09.002","url":null,"abstract":"<div><div>Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points <span><math><mi>q</mi><mi>α</mi></math></span> with <span><math><mi>α</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and <span><math><mi>q</mi><mo>∈</mo><mi>Z</mi></math></span>. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of <strong><em>α</em></strong> whose accumulation points are all of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In the second part we focus primarily on the case when the coordinates of <strong><em>α</em></strong> together with 1 form a basis for an algebraic number field <em>K</em>. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>, this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in <em>K</em>) of a single ellipse, or of a pair of hyperbolas, depending on whether or not <em>K</em> has a non-trivial embedding into <span><math><mi>C</mi></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 1-38"},"PeriodicalIF":0.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699240","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}
Pub Date : 2024-11-05DOI: 10.1016/j.jnt.2024.09.006
Florian Luca , István Pink
In this paper, we find all the solutions of the Diophantine equation from the title.
在本文中，我们将从题目中找出 Diophantine 方程的所有解。
{"title":"On the Diophantine equation 2s + pk = m2 with a Fermat prime p","authors":"Florian Luca , István Pink","doi":"10.1016/j.jnt.2024.09.006","DOIUrl":"10.1016/j.jnt.2024.09.006","url":null,"abstract":"<div><div>In this paper, we find all the solutions of the Diophantine equation from the title.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 49-71"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699242","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}
Pub Date : 2024-10-31DOI: 10.1016/j.jnt.2024.09.003
Pratiksha Shingavekar
Given a sixth power free integer a, let be the elliptic curve defined by . We prove explicit results for the lower density of sixth power free integers a for which the 3-isogeny induced Selmer group of over has dimension ≤1. The results are proven by refining the strategy of Davenport–Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for 3-isogeny induced Selmer groups.
给定一个六次幂自由整数 a，设 Ea 是由 y2=x3+a 定义的椭圆曲线。我们证明了六次幂自由整数 a 的低密度的明确结果，对于这些低密度的六次幂自由整数 a，Ea 在 Q(μ3) 上的 3-isogeny induced Selmer 群的维数≤1。这些结果是通过改进达文波特-海尔布隆的策略，将积分二元三次形式的统计量与 3-isogeny induced Selmer 群的统计量联系起来而证明的。
{"title":"Statistics for 3-isogeny induced Selmer groups of elliptic curves","authors":"Pratiksha Shingavekar","doi":"10.1016/j.jnt.2024.09.003","DOIUrl":"10.1016/j.jnt.2024.09.003","url":null,"abstract":"<div><div>Given a sixth power free integer <em>a</em>, let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> be the elliptic curve defined by <span><math><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>a</mi></math></span>. We prove explicit results for the lower density of sixth power free integers <em>a</em> for which the 3-isogeny induced Selmer group of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> over <span><math><mi>Q</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> has dimension ≤1. The results are proven by refining the strategy of Davenport–Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for 3-isogeny induced Selmer groups.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 72-94"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699243","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}
Pub Date : 2024-10-31DOI: 10.1016/j.jnt.2024.09.007
Sonja Žunar
Using Poincaré series of K-finite matrix coefficients of integrable antiholomorphic discrete series representations of , we construct a spanning set for the space of Siegel cusp forms of weight ρ for Γ, where ρ is an irreducible polynomial representation of of highest weight with , and Γ is a discrete subgroup of commensurable with . Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.
{"title":"Construction and non-vanishing of a family of vector-valued Siegel Poincaré series","authors":"Sonja Žunar","doi":"10.1016/j.jnt.2024.09.007","DOIUrl":"10.1016/j.jnt.2024.09.007","url":null,"abstract":"<div><div>Using Poincaré series of <em>K</em>-finite matrix coefficients of integrable antiholomorphic discrete series representations of <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, we construct a spanning set for the space <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of Siegel cusp forms of weight <em>ρ</em> for Γ, where <em>ρ</em> is an irreducible polynomial representation of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> of highest weight <span><math><mi>ω</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>…</mo><mo>≥</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>></mo><mn>2</mn><mi>n</mi></math></span>, and Γ is a discrete subgroup of <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> commensurable with <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 95-123"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699244","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}
Pub Date : 2024-10-31DOI: 10.1016/j.jnt.2024.09.004
Vjekoslav Kovač
Erdős and Graham found it conceivable that the best n-term Egyptian underapproximation of almost every positive number for sufficiently large n gets constructed in a greedy manner, i.e., from the best -term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero.
厄尔多斯和格雷厄姆发现，几乎每一个足够大 n 的正数的最佳 n 项埃及下逼近都可以用贪婪的方式构造出来，即从最佳 (n-1)- 项埃及下逼近中构造出来。我们的研究表明，事实恰恰相反：具有这种性质的实数集合的勒贝格度量为零。
{"title":"On eventually greedy best underapproximations by Egyptian fractions","authors":"Vjekoslav Kovač","doi":"10.1016/j.jnt.2024.09.004","DOIUrl":"10.1016/j.jnt.2024.09.004","url":null,"abstract":"<div><div>Erdős and Graham found it conceivable that the best <em>n</em>-term Egyptian underapproximation of almost every positive number for sufficiently large <em>n</em> gets constructed in a greedy manner, i.e., from the best <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 39-48"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699241","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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