Pub Date : 2023-09-01DOI: 10.1017/s0013091523000482
A. Gauvan
We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $left{X_k : k geq 1right}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions begin{equation*}Omega_{text{rand},text{lin}} := left{ frac{pi X_k}{k}: k geq 1right}end{equation*} and begin{equation*}Omega_{text{rand},text{lac}} := left{frac{pi X_k}{2^k} : kgeq 1 right}.end{equation*} We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({mathbb{R}}^2)$ for any $1 lt p lt infty$ .
{"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}
Pub Date : 2023-08-01DOI: 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}
Pub Date : 2023-08-01DOI: 10.1017/S0013091523000421
Jin-Hui Fang
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.
{"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}
Pub Date : 2023-08-01DOI: 10.1017/S0013091523000457
Olivia X. M. Yao
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.
{"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}
Pub Date : 2023-08-01DOI: 10.1017/S0013091523000469
Jingcheng Liu, Min-Wei Tang, Shan Wu
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.
{"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}
Pub Date : 2023-08-01DOI: 10.1017/s0013091523000470
Allan P. Donsig, Adam H. Fuller, David R. Pitts
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.
{"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}
Pub Date : 2023-08-01DOI: 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}
Pub Date : 2023-08-01DOI: 10.1017/S0013091523000433
A. Dubickas
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.
{"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}
Pub Date : 2023-06-30DOI: 10.1017/S0013091523000299
Niamh Farrell, Caroline Lassueur
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.
{"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}
Pub Date : 2023-05-25DOI: 10.1017/S0013091523000287
Ugo Bessi
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.
{"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}
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