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Perron’s capacity of random sets 随机集的Perron容量
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-09-01 DOI: 10.1017/s0013091523000482
A. Gauvan
{"title":"Perron’s capacity of random sets","authors":"A. Gauvan","doi":"10.1017/s0013091523000482","DOIUrl":"https://doi.org/10.1017/s0013091523000482","url":null,"abstract":"\u0000 We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables \u0000 \u0000 \u0000 $left{X_k : k geq 1right}$\u0000 \u0000 uniformly distributed in \u0000 \u0000 \u0000 $(0,1)$\u0000 \u0000 and independent, we consider the following random sets of directions\u0000\u0000 \u0000 \u0000 begin{equation*}Omega_{text{rand},text{lin}} := left{ frac{pi X_k}{k}: k geq 1right}end{equation*}\u0000 \u0000 and\u0000\u0000 \u0000 \u0000 begin{equation*}Omega_{text{rand},text{lac}} := left{frac{pi X_k}{2^k} : kgeq 1 right}.end{equation*}\u0000 \u0000 \u0000 We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on \u0000 \u0000 \u0000 $L^p({mathbb{R}}^2)$\u0000 \u0000 for any \u0000 \u0000 \u0000 $1 lt p lt infty$\u0000 \u0000 .","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41978588","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}
引用次数: 0
PEM series 2 volume 66 issue 3 Cover and Back matter PEM系列2卷66期3封面和封底
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-08-01 DOI: 10.1017/s0013091523000524
{"title":"PEM series 2 volume 66 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0013091523000524","DOIUrl":"https://doi.org/10.1017/s0013091523000524","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":" ","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42257992","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}
引用次数: 0
On the density of bounded bases 关于有界基的密度
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-08-01 DOI: 10.1017/S0013091523000421
Jin-Hui Fang
{"title":"On the density of bounded bases","authors":"Jin-Hui Fang","doi":"10.1017/S0013091523000421","DOIUrl":"https://doi.org/10.1017/S0013091523000421","url":null,"abstract":"Abstract For a nonempty set A of integers and an integer n, let $r_{A}(n)$ be the number of representations of n in the form $n=a+a'$, where $aleqslant a'$ and $a, a'in A$, and $d_{A}(n)$ be the number of representations of n in the form $n=a-a'$, where $a, a'in A$. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion of n, i.e., $n=sum_{iin S(n)} 2^i$, $S(-n)=-S(n)$ and $S(0)=emptyset$. For real number x, let $A(-x,x)$ be the number of elements $ain A$ with $-xleqslant aleqslant x$. The famous Erdős-Turán Conjecture states that if A is a set of positive integers such that $r_A(n)geqslant 1$ for all sufficiently large n, then $limsup_{nrightarrowinfty}r_A(n)=infty$. In 2004, Nešetřil and Serra initially introduced the notation of “bounded” property and confirmed the Erdős-Turán conjecture for a class of bounded bases. They also proved that, there exists a set A of integers satisfying $r_A(n)=1$ for all integers n and $|S(x)bigcup S(y)|leqslant 4|S(x+y)|$ for $x,yin A$. On the other hand, Nathanson proved that there exists a set A of integers such that $r_A(n)=1$ for all integers n and $2log x/log 5+c_1leqslant A(-x,x)leqslant 2log x/log 3+c_2$ for all $xgeqslant 1$, where $c_1,c_2$ are absolute constants. In this paper, following these results, we prove that, there exists a set A of integers such that: $r_A(n)=1$ for all integers n and $d_A(n)=1$ for all positive integers n, $|S(x)bigcup S(y)|leqslant 4|S(x+y)|$ for $x,yin A$ and $A(-x,x) gt (4/log 5)loglog x+c$ for all $xgeqslant 1$, where c is an absolute constant. Furthermore, we also construct a family of arbitrarily spare such sets A.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"832 - 844"},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42559236","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}
引用次数: 0
New mock theta functions and formulas for basic hypergeometric series 新的模拟函数和基本超几何级数的公式
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-08-01 DOI: 10.1017/S0013091523000457
Olivia X. M. Yao
{"title":"New mock theta functions and formulas for basic hypergeometric series","authors":"Olivia X. M. Yao","doi":"10.1017/S0013091523000457","DOIUrl":"https://doi.org/10.1017/S0013091523000457","url":null,"abstract":"Abstract In recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"868 - 896"},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56897244","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}
引用次数: 0
The spectral eigenmatrix problems of planar self-affine measures with four digits 平面四位数自仿射测度的谱特征矩阵问题
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-08-01 DOI: 10.1017/S0013091523000469
Jingcheng Liu, Min-Wei Tang, Shan Wu
{"title":"The spectral eigenmatrix problems of planar self-affine measures with four digits","authors":"Jingcheng Liu, Min-Wei Tang, Shan Wu","doi":"10.1017/S0013091523000469","DOIUrl":"https://doi.org/10.1017/S0013091523000469","url":null,"abstract":"Abstract Given a Borel probability measure µ on $mathbb{R}^n$ and a real matrix $Rin M_n(mathbb{R})$. We call R a spectral eigenmatrix of the measure µ if there exists a countable set $Lambdasubset mathbb{R}^n$ such that the sets $E_Lambda=big{{rm e}^{2pi i langlelambda,xrangle}:lambdain Lambdabig}$ and $E_{RLambda}=big{{rm e}^{2pi i langle Rlambda,xrangle}:lambdain Lambdabig}$ are both orthonormal bases for the Hilbert space $L^2(mu)$. In this paper, we study the structure of spectral eigenmatrix of the planar self-affine measure $mu_{M,D}$ generated by an expanding integer matrix $Min M_2(2mathbb{Z})$ and the four-elements digit set $D = {(0,0)^t,(1,0)^t,(0,1)^t,(-1,-1)^t}$. Some sufficient and/or necessary conditions for R to be a spectral eigenmatrix of $mu_{M,D}$ are given.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"897 - 918"},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43000130","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}
引用次数: 1
Corrigendum: Von Neumann Algebras and Extensions of Inverse Semigroups 勘误:Von Neumann代数和逆半群的扩展
3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-08-01 DOI: 10.1017/s0013091523000470
Allan P. Donsig, Adam H. Fuller, David R. Pitts
{"title":"Corrigendum: Von Neumann Algebras and Extensions of Inverse Semigroups","authors":"Allan P. Donsig, Adam H. Fuller, David R. Pitts","doi":"10.1017/s0013091523000470","DOIUrl":"https://doi.org/10.1017/s0013091523000470","url":null,"abstract":"An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the ‘Save PDF’ action button.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"75 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136222850","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}
引用次数: 0
PEM series 2 volume 66 issue 3 Cover and Front matter PEM系列2第66卷第3期封面和封面
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-08-01 DOI: 10.1017/s0013091523000512
{"title":"PEM series 2 volume 66 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s0013091523000512","DOIUrl":"https://doi.org/10.1017/s0013091523000512","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":" ","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43749073","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}
引用次数: 0
Every Salem number is a difference of two Pisot numbers 每个塞勒姆数都是两个皮索数的差
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-08-01 DOI: 10.1017/S0013091523000433
A. Dubickas
{"title":"Every Salem number is a difference of two Pisot numbers","authors":"A. Dubickas","doi":"10.1017/S0013091523000433","DOIUrl":"https://doi.org/10.1017/S0013091523000433","url":null,"abstract":"Abstract In this note, we prove that every Salem number is expressible as a difference of two Pisot numbers. More precisely, we show that for each Salem number α of degree d, there are infinitely many positive integers n for which $alpha^{2n-1}-alpha^n+alpha$ and $alpha^{2n-1}-alpha^n$ are both Pisot numbers of degree d and that the smallest such n is at most $6^{d/2-1}+1$. We also prove that every real positive algebraic number can be expressed as a quotient of two Pisot numbers. Earlier, Salem himself had proved that every Salem number can be written in this way.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"862 - 867"},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47410861","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}
引用次数: 0
Trivial source character tables of $operatorname{SL}_2(q)$, part II $operatorname{SL}_2(q)$的平凡源字符表，第二部分
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-06-30 DOI: 10.1017/S0013091523000299
Niamh Farrell, Caroline Lassueur
{"title":"Trivial source character tables of $operatorname{SL}_2(q)$, part II","authors":"Niamh Farrell, Caroline Lassueur","doi":"10.1017/S0013091523000299","DOIUrl":"https://doi.org/10.1017/S0013091523000299","url":null,"abstract":"Abstract We compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups $operatorname{SL}_{2}(q)$ for q even over a large enough field of odd characteristics. This article is a continuation of our article Trivial Source Character Tables of $operatorname{SL}_{2}(q)$, where we considered, in particular, the case in which q is odd in non-defining characteristic.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"689 - 709"},"PeriodicalIF":0.7,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43956902","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}
引用次数: 1
Counting periodic orbits on fractals weighted by their Lyapounov exponents 用Lyapounov指数加权分形的周期轨道计数
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2023-05-25 DOI: 10.1017/S0013091523000287
Ugo Bessi
{"title":"Counting periodic orbits on fractals weighted by their Lyapounov exponents","authors":"Ugo Bessi","doi":"10.1017/S0013091523000287","DOIUrl":"https://doi.org/10.1017/S0013091523000287","url":null,"abstract":"Abstract Several authors have shown that Kusuoka’s measure κ on fractals is a scalar Gibbs measure; in particular, it maximizes a pressure. There is also a different approach, in which one defines a matrix-valued Gibbs measure µ, which induces both Kusuoka’s measure κ and Kusuoka’s bilinear form. In the first part of the paper, we show that one can define a ‘pressure’ for matrix-valued measures; this pressure is maximized by µ. In the second part, we use the matrix-valued Gibbs measure µ to count periodic orbits on fractals, weighted by their Lyapounov exponents.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"710 - 757"},"PeriodicalIF":0.7,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49517558","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}
引用次数: 0
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