{"title":"Fuglede's conjecture on cyclic groups of order pnq","authors":"R. Malikiosis, M. N. Kolountzakis","doi":"10.19086/da.2071","DOIUrl":"https://doi.org/10.19086/da.2071","url":null,"abstract":"Fuglede's conjecture on cyclic groups of order $p^nq$, Discrete Analysis 2017:12, 16 pp.\u0000\u0000A conjecture of Fuglede from 1974 states that a measurable set $Esubset mathbb R^n$ of positive Lebesgue measure tiles $mathbb R^n$ by translations if and only if the space $L^2(E)$ admits an orthonormal basis of exponential functions ${ e^{2pi i lambdacdot x}: lambdainLambda}$. (The set $Lambda$ is called a _spectrum_ for $E$.)\u0000\u0000We now know that the conjecture is false in dimensions 3 and higher, with counterexamples due to Tao, Kolountzakis, Matolcsi, Farkas, Revesz, and Mora. Nonetheless, there are important special cases where the conjecture has been confirmed, such as convex bodies in $mathbb R^2$ (due to Iosevich, Katz and Tao), and the case where either the spectrum or the translation set is a lattice (Fuglede). Research on Fuglede's conjecture has also established a broader family of correspondences between geometric and harmonic-analytic properties of sets that are of interest in their own right.\u0000\u0000For general non-convex sets in dimensions 1 and 2, the conjecture remains open in both directions. The main focus so far has been on the corresponding discrete problems and their links to questions in combinatorial number theory and factorization of Abelian groups. On one hand, the \"tiling implies spectrum\" part in dimension 1 would follow from an independently made conjecture of Coven and Meyerovitz on characterizing finite sets that tile the integers by translations. On the other hand, given that the higher-dimensional counterexamples are based on adapting finite fields constructions to the continuous setting, it is tempting to try to disprove the conjecture in lower dimensions in a similar manner.\u0000\u0000In this paper, the authors prove that the conjecture is true in both directions in cyclic groups of order $N=p^nq$, where $p$ and $q$ are distinct primes. The \"tiling implies spectrum\" direction follows immediately from the earlier work of Coven-Meyerowitz and Łaba. The main new contribution is the resolution of the more difficult \"spectrum implies tiling\" question in the case under consideration. \u0000\u0000The proof is based on the existing results on the structure of vanishing sums of roots of unity. This is relevant because if $Asubset mathbb Z_N$ is spectral, then the orthogonality relations between the exponential functions in the spectrum can be expressed in terms of the zeros of the _mask polynomial_ $A(x)=sum_{ain A} x^a$ on the unit circle. Lam and Leung proved that any vanishing sum of roots of unity of order $p^n q^m$ can be expressed as a linear combination of (roots of unity forming) rotated regular $p$-gons and $q$-gons, with positive coefficients. In the present paper, Kolountzakis and Malikiokis use this to analyze the structure of spectral sets $A$ in $mathbb Z_{p^nq}$, proving that such sets must satisfy the Coven-Meyerowitz tiling conditions and therefore must tile the group. \u0000\u0000The use of the Lam-Leung theorem is an exciting new development in this","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68392974","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}
{"title":"Proof of a conjecture of Kleinberg-Sawin-Speyer","authors":"Luke Pebody","doi":"10.19086/DA.3733","DOIUrl":"https://doi.org/10.19086/DA.3733","url":null,"abstract":"In Ellenberg and Gijswijt's groundbreaking work~cite{EllenbergGijswijt}, the authors show that a subset of $mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must be of size at most $2.755^n$ (no prior upper bound was known of $(3-epsilon)^n)$), and provide for any prime $p$ a value $lambda_p<p$ such that any subset of $mathbb{Z}_p^{n}$ with no arithmetic progression of length 3 must be of size at most $lambda_p^n$. Blasiak et al~cite{BlasiakEtAl} showed that the same bounds apply to tri-coloured sum-free sets, which are triples ${(a_i,b_i,c_i):a_i,b_i,c_iinmathbb{Z}_p^{n}}$ with $a_i+b_j+c_k=0$ if and only if $i=j=k$. Building on this work, Kleinberg, Sawin and Speyer~cite{KleinbergSawinSpeyer} gave a description of a value $mu_p$ such that no tri-coloured sum-free sets of size $e^{mu_p n}$ exist in $mathbb{Z}_p^{n}$, but for any $epsilon>0$, such sets of size $e^{(mu_p-epsilon) n}$ exist for all sufficiently large $n$. The value of $mu_p$ was left open, but a conjecture was stated which would imply that $e^{mu_p}=lambda_p$, i.e. the Ellenberg-Gijswijt bound is correct for the sum-free set problem. The purpose of this note is to close that gap. The conjecture of Kleinberg, Sawin and Speyer is true, and the Ellenberg-Gijswijt bound is the correct exponent for the sum-free set problem.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68393082","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}
{"title":"Approximate invariance for ergodic actions of amenable groups.","authors":"M. Bjorklund, A. Fish","doi":"10.19086/DA.8471","DOIUrl":"https://doi.org/10.19086/DA.8471","url":null,"abstract":"We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups. \u0000As an application of these techniques, we prove a dynamical generalization of Kneser's celebrated density theorem for subsets in $(bZ,+)$, valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68393375","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}
{"title":"The growth rate of tri-colored sum-free sets","authors":"Robert D. Kleinberg, W. Sawin, David E. Speyer","doi":"10.19086/da.3734","DOIUrl":"https://doi.org/10.19086/da.3734","url":null,"abstract":"The growth rate of tri-colored sum-free sets, Discrete Analysis 2018:12, 10 pp.\u0000\u0000This paper contributes to the remarkable collection of results that followed in the wake of the 2016 breakthrough by Ellenberg and Gijswijt on the cap set problem, which asks for the maximal size of a 3-term progression-free subset of $mathbb{F}_3^n$. The polynomial method of Ellenberg and Gijswijt, who followed the lead of Croot, Lev, and Pach after whom the method is now named, showed for the first time that the size of such a set is bounded by a polynomial in the size of the ambient space. More specifically, they showed that a cap set in $mathbb{F}_3^n$ can be of size at most $(2.756)^n$.\u0000\u0000This paper considers a variant of the cap set problem, namely the question of how large a tri-coloured sum-free subset of $mathbb{F}_3^n$ can be. By a tri-coloured sum-free subset we mean a collection of triples $(a_i,b_i,c_i)$ in $(mathbb{F}_3^n)^3$ such that $a_i+b_j+c_k=0$ if and only if $i=j=k$. Note that a cap set $Asubseteq mathbb{F}_3^n$ gives rise to a tri-coloured sum-free set, namely the collection of triples ${(a,a,a): ain A}$. \u0000\u0000It is therefore not entirely surprising that the Croot-Lev-Pach polynomial method can be used to show that if $q$ is a prime power, then a tri-coloured sum-free set in $mathbb{F}_q^n$ can have size at most $3theta^n$, where $theta$ is the solution to an explicit optimisation problem. This is one of the results appearing in the paper [\"On cap sets and the group-theoretic approach to matrix multiplication\"](http://discreteanalysisjournal.com/article/1245), by Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans, which Discrete Analysis published earlier this year.\u0000\u0000In the present paper the authors show this bound to be tight to within a subexponential factor. The lower bound is based on a construction of an $S_3$-symmetric probability distribution on ${(a,b,c)inmathbb{Z}_{geq 0}^3:a+b+c=n}$ such that its marginal achieves the maximum entropy among all probability distributions on ${0,1,…,n}$ with mean $n/3$. In answer to a conjecture which was posed in the original preprint version of this article, the existence of such a distribution was established by Pebody, whose [article](http://discreteanalysisjournal.com/article/3733-proof-of-a-conjecture-of-kleinberg-sawin-speyer) is being published in Discrete Analysis alongside the present paper.\u0000\u0000The question of whether the Croot-Lev-Pach polynomial method also yields a tight bound for the cap set problem remains open.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68392643","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}
{"title":"Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics","authors":"Jan Hkazla, Thomas Holenstein, Elchanan Mossel","doi":"10.19086/da.6513","DOIUrl":"https://doi.org/10.19086/da.6513","url":null,"abstract":"Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics, Discrete Analysis 2018:20, 63 pp.\u0000\u0000Szemeredi's theorem states that for every positive integer $ell$ and every $mu>0$ there exists $N$ such that every subset of ${1,2,dots,N}$ of density at least $mu$ contains an arithmetic progression of length $ell$. It is not hard to prove that this is equivalent to the statement that if $N$ is sufficiently large, then every function $f:mathbb Z_Nto[0,1]$ (where $mathbb Z_N$ is the cyclic group of order $N$) with average value at least $mu$ satisfies an inequality of the form\u0000\u0000$$mathbb E_{x,d}f(x)f(x+d)dots f(x+(ell-1)d)geq delta$$\u0000\u0000where $delta$ is a positive constant that depends only on $mu$ and $ell$. \u0000\u0000We can express this as a statement about a product of random variables, as follows. Let $x$ and $d$ be chosen uniformly at random and for each $1leq ileqell$ let $X^{(i)}$ be the random variable that takes the value $x+(i-1)d$. Then for every function $f:mathbb Z_N$ that takes values in $[0,1]$, if $mathbb E[f(X^{(i)})]geqmu$ for each $i$, then we also have that $mathbb E f(X^{(1)})dots f(X^{(ell)})geqdelta$. A sequence of random variables with this property is called _same-set hitting_. Note that the random variables $X^{(i)}$ here are not independent, but they are individually uniformly distributed. In particular, they are identically distributed.\u0000\u0000The equivalent form of Szemeredi's theorem given above can be stated and proved for other Abelian groups. A particularly well-known case is when the group is $mathbb F_p^n$ for some fixed prime $pgeqell$. Here we can say more about the corresponding random variables $X^{(i)}$. Now they take values in $mathbb F_p^n$, and if for each coordinate $j$ we form the vector $underline{X}_j=(X^{(1)}_j,dots,X^{(ell)}_j)$, we find that the vectors $underline{X}_j$ are independent and identically distributed. Indeed, each one is a random (possibly degenerate) arithmetic progression in the group $mathbb F_p$. \u0000\u0000The purpose of this paper is to consider this situation in general, and in particular to try to understand which systems of random variables $X^{(1)},dots,X^{(ell)}$ satisfying the above conditions are same-set hitting.\u0000\u0000A related concept, which also comes up in additive combinatorics, is that of being _set hitting_. This is the natural off-diagonal strengthening of being same-set hitting. That is, the random variables are set hitting if whenever $f_1,dots,f_ell$ are functions taking values in $[0,1]$ and $mathbb Ef_i(X^{(i)})geqmu$ for each $i$, we have that $mathbb Ef_1(X^{(1)})dots f_ell(X^{(ell)})geqdelta$, where once again $delta$ depends only on $mu$ and $ell$.\u0000\u0000To see that this is a considerably stronger property, one need only look at the case of random arithmetic progressions $(X^{(1)},X^{(2)},X^{(3)})$ of length 3 in $mathbb F_3^n$. If we let $f_1=f_2(x)=1$ if $x_1=0$ and $0$ otherwise, and $f_3(x)=1$ if $x_1=1$ and $0$ otherwise, then each $mathbb E f_i(X^{(i)})$ is equa","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2015-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68392825","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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