Zarko Mijajlovic, Milos Milosevic, Aleksandar Perovic
In this paper we introduce the notion of the $P$-sequences and apply their properties in studying representability of real numbers. Another application of $P$-sequences we find in generating the Prouhet-Tarry-Escott pairs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In a recent note W. Kohnen asks whether the values of Dedekind sums are dense in the field of $p$-adic numbers. The present paper answers this question. Dedekind sums do not approximate units of $\mathbb Z_2$ or $\mathbb Z_3$, so they are not dense in $\mathbb Q_2$ or $\mathbb Q_3$. But they are dense in $\mathbb Q_p$ if $p\ge 5$.
Triangular and tetrahedral number differences of sumset sizes in additive number theory
本文研究有限整数集合sumset大小的完整范围，发现在大小为4的集合流行sumset大小分布中存在与三角数和四面体数相关的意外模式。
We obtain lower bounds for the cardinality of $k$-fold sum-sets of reciprocals of elements of suitable defined short intervals in high degree extensions of finite fields. Combining our results with bounds for multilinear character sums we obtain new results on incomplete multilinear Kloosterman sums in finite fields.
Decomposition of real numbers into sums of Lüroth sets
本文研究了实数分解为Lüroth集之和的性质，探讨了其模1同余性和Hausdorff维度，并将连分数经典结果扩展至Lüroth展开领域。
We show that a random set of integers with density 0 has almost always more differences than sums. This proves a conjecture by Martin and O'Bryant.
Let $k \ge 2$ be an integer. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called Sidon set if all the two terms sums formed by the elements of $A$ are different. Many years ago P. Erdős, A. Sárközy and V. T. Sós asked whether there exists a Sidon set which is asymptotic basis of order $3$. In this paper we prove the existence of a Sidon set $A$ with positive lower density of the three fold sumset $A + A + A$ by using probabilistic methods.
Inverse problems for sumset sizes of finite sets of integers
本文研究了有限整数集的和集大小序列的反问题，探讨了不同集之间的关系及增长速率。
We obtain quantitative estimates for the asymptotic density of subsets of the two-dimensional integer lattice which contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov's mean value theorem applicable to binary forms.
We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to classical character sum bounds of Polya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements.
We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup $Γ_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum functi…
Let $h$ be a positive integer and let $\varepsilon > 0$. The Haight-Ruzsa method produces a positive integer $m^*$ and a subset $A$ of the additive abelian group $\mathbf{Z}/m^*\mathbf{Z}$ such that the difference set is large in the sense that $A-A = \mathbf{Z}/m^*\mathbf{Z}$ and $h$-fold sumset is small in the sense that $|hA| < \varepsilon m^*$. This note describes, and in a modest way extends, the Haight-Ruzsa argument, and constructs sets with more differences than multiple sums in other additive abelian groups.
We discuss the problem of finding distinct integer sets $\{x_1,x_2,...,x_n\}$ where each sum $x_i+x_j, i \ne j$ is a square, and $n \le 7$. We confirm minimal results of Lagrange and Nicolas for $n=5$ and for the related problem with triples. We provide new solution sets for $n=6$ to add to the single known set. This provides new information for problem D15 in Guy's {\it Unsolved Problems in Number Theory}
For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain…
We prove new general results on sumsets of sets having Szemerédi--Trotter type. This family includes convex sets, sets with small multiplicative doubling, images of sets under convex/concave maps and others.
Proofs of Mizuno's Conjectures on Generalized Rank Two Nahm Sums
本研究针对Mizuno提出的14组秩二广义模Nahm和候选集合，通过构造相关Rogers-Ramanujan型恒等式证明了其中8组的模性，给出剩余2组的猜想公式，同时证明了相关向量值函数的模变换猜想，并发现若干新的非模恒等式。
We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the absolute value of the character sum over $U$ and any subset $C$ (which satisfies some restrictions on its size $|C|$) can be improved. Examples of such suitable self-maps are inversion and squaring. Then we apply this new bound to trace products and sum-product equations and improve results of the first author and of Gyarmati and Sárközy.
A More Sums Than Differences (MSTD) set is a set of integers A contained in {0, ..., n-1} whose sumset A+A is larger than its difference set A-A. While it is known that as n tends to infinity a positive percentage of subsets of {0, ..., n-1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family's density among subsets of {0, ..., n-1} is at least C/n^4 for some C>0, significan…
In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.
Let $1<q<2$ and \[ Λ(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $Λ(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $Λ(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $α$ such that $q|α|<1$, then $Λ(q)$ is dense in $\mathbb{R}$.
We investigate here sums of triangular numbers $f(x):=\sum_i b_i T_{x_i}$ where $T_n$ is the $n$-th triangular number. We show that for a set of positive integers $S$ there is a finite subset $S_0$ such that $f$ represents $S$ if and only if $f$ represents $S_0$. However, computationally determining $S_0$ is ineffective for many choices of $S$. We give an explicit and efficient algorithm to determine the set $S_0$ under certain Generalized Riemann Hypotheses, and implement the algorithm to determine $S_0$ when $S$ is the set of all odd integers.
R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A+B must be piecewise Bohr. We generalize the latter result to cases where A has Banach density 0, giving a new proof of the previous results in the process.
Exponential sums with polynomials and their applications to primes in sparse sets
本文研究带多项式的指数和，证明新上界，并应用于改进 Piatetski-Shapiro 序列及迭代序列中素数渐近公式的允许范围。
We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set $A$ the maximal size of a set whose difference set avoids $A$ will be related to positive exponential sums using frequencies from $A$.
Compression and complexity for sumset sizes in additive number theory
本文研究了加法数论中和集大小的压缩与复杂性，探讨了h-fold和集的几何与计算复杂性，提出了一种构造直径小的和集A'的方法。
Let $A$ be a subset of $\mathbb{Z} / N\mathbb{Z}$ and let $\mathcal{R}$ be the set of large Fourier coefficients of $A$. Properties of $\mathcal{R}$ have been studied in works of M.-- C. Chang, B. Green and the author. In the paper we obtain some new results on sets of large exponential sums.
Sums and differences of sets (improvement over AlphaEvolve)
本文针对集合的和与差问题，通过显式构造包含超过10^43546个元素的U集合，将下界从1.1584改进至1.173050，优于AlphaEvolve的结果。
In the paper we are studying some properties of subsets Q of sums of dissociated sets. The exact upper bound for the number of solutions of the following equation (1) q_1 + ... + q_p = q_{p+1} + ... + q_{2p}, q_i \in Q in groups F_2^n is found. Using our approach, we easily prove a recent result of J. Bourgain on sets of large exponential sums and obtain a tiny improvement of his theorem. Besides an inverse problem is considered in the article. Let Q be a set belonging a sumset of two dissociated sets such that equation (1) has many solutions. We prove that in the case the large proportion of …
We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical object of study of additive combinatorics.
It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1/\sqrt{\log\log\log N}$. The proof relies on ideas from classical work of Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.
In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White [24] and several of subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.
Given a subset of real numbers $A$ with small product $AA$ we obtain a new upper bound for the additive energy of $A$. The proof uses a natural observation that level sets of convolutions of the characteristic function of $A$ have small product with $A$.
A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set $A\subset \mathbb{Z}$ such that $|A+A|<|A-A|$. Though it was believed that the percentage of subsets of $\{0,...,n\}$ that are sum-dominant tends to zero, in 2006 Martin and O'Bryant \cite{MO} proved a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that $|ε_1A+...+ε_kA|>|δ_1A+...+δ_kA|$ a positive percent of the time for all nontrivial choices of $ε_j,δ_j\in \{-1,1\}$. Previous approaches proved the existence of infinitely many …
Let A be a finite subset of the integers or, more generally, of any abelian group, written additively. The set A has "more sums than differences" if |A+A|>|A-A|. A set with this property is called an MSTD set. This paper gives explicit constructions of families of MSTD sets of integers.
Elliptic Dedekind sums were introduced by R. Sczech as generalizations of classical Dedekind sums to complex lattices. We show that for any lattice with real $j$-invariant, the values of suitably normalized elliptic Dedekind sums are dense in the real numbers. This extends an earlier result of Ito for Euclidean imaginary quadratic rings. Our proof is an adaptation of the recent work of Kohnen, which gives a new proof of the density of values of classical Dedekind sums.
If $P$ is a lattice polytope (that is, the convex hull of a finite set of lattice points in $\mathbf{R}^n$), then every sum of $h$ lattice points in $P$ is a lattice point in the $h$-fold sumset $hP$. However, a lattice point in the $h$-fold sumset $hP$ is not necessarily the sum of $h$ lattice points in $P$. It is proved that if the polytope $P$ is a union of unimodular simplices, then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.
Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leqα\leqβ\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that $\mathrm{d}(A)=α$ and $\mathrm{d}(A+A)=β$. More generally we study the set of $k$-tuples $(\mathrm{d}(iA))_{1\leq i\leq k}$ for $A\subset \mathbb{N}$. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of ``pseudo $s$th powers''. We consider similar problems for subsets of the circle $\mathbb{R}/\mathbb{Z}$, …
Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a sum of $h$ elements of $\mathcal{A}$. The structure of the set $(h\mathcal{A})^{(t)}$ is completely determined for all $h \geq h_t$.
We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets. Thirdly we show that for every fixed ordered pair of non-negative integers (j,k), as n -> \infty a positive proportion of the subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.
We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are $(2^d-1, 2^{d-1}-1, 2^{d-2}-1)$ cyclic difference sets in the multiplicative group of the finite field $F_{2^d}$ of $2^d$ elements, with $d \geq 2$. We show that, except for a few cases with small $d$, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ra…
We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$. These bounds rely on some recent advances in additive combinatorics.
Metric results on sumsets and Cartesian products of classes of Diophantine sets
受Erdős关于刘维尔数和积结果启发，研究指定无理性指数实数类和集的度量性质，证明其和集勒贝格测度下几乎覆盖全体实数，补集豪斯多夫维数小，还拓展到笛卡尔积维数、缺数字康托尔集填充维数等问题，推广了Erdős的方法。
Sums of singular series with large sets and the tail of the distribution of primes
本文研究k相对于h较大时区间内集合的奇异级数常数平均值，将其应用于素数分布尾部，在Hardy--Littlewood猜想下得出含至少k个素数的对数长度区间数量的上界估计。
Sums and differences of sets: a further improvement over AlphaEvolve
本文提出集合和差问题的新进展，通过构造 $U$ 集合序列，在极限下确立了 $θ= 1.173077$ 的新下界，改进了 AlphaEvolve 及后续显式构造的近期结果。
Given a prime $p$, an integer $H\in[1,p)$, and an arbitrary set $\cal M\subseteq \mathbb F_p^*$, where $\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\cal M)$ denote the number of solutions to the congruence $$ xm\equiv yn\bmod p $$ for which $x,y\in[1,H]$ and $m,n\in\cal M$. In this paper, we bound $J(H,\cal M)$ in terms of $p$, $H$ and the cardinality of $\cal M$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results…
Difference sets and positive exponential sums II: cubic residues in cyclic groups
本文通过构造非负指数和，给出了循环群中差集避免三次剩余的集合基数的上界。
A sum-dominant set is a finite set $A$ of integers such that $|A+A| > |A-A|$. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sum-dominant subsets of $\{0,\dots,n\}$ is bounded below by a positive constant as $n\to\infty$. Hegarty then extended their work and showed that for any prescribed $s,d\in\mathbb{N}_0$, the proportion $ρ^{s,d}_n$ of subsets of $\{0,\dots,n\}$ that are missing exactly $s$ sums in $\{0,\dots,2n\}$ and exactly $2d$ differences in…
In this paper we show that if $A$ is a subset of the primes with positive relative density $δ$, then $A+A$ must have positive upper density $C_1δe^{-C_2(\log(1/δ))^{2/3}(\log\log(1/δ))^{1/3}}$ in $\mathbb{N}$. Our argument applies the techniques developed by Green and Green-Tao used to find arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo $M$.
Let $\mathcal{A}$ and $\mathcal{B}$ be sets of polynomials of degree $n$ over a finite field. We show, that if $\mathcal{A}$ and $\mathcal{B}$ are large enough, then $A+B$ has an irreducible divisor of large degree for some $A\in\mathcal{A}$ and $B\in \mathcal{B}$.
We improve a special case of the Lam-Leung lower bound on the number of elements in a vanishing sum of $N$-th roots of unity. Using this result, we extend the Favard length estimates due to Bond, Łaba, and Volberg to a new class of rational product Cantor sets in $\mathbb{R}^2$.
For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ be its continued fraction expansion. Let $s_n(x)=\sum_{j=1}^n a_j(x)$. The Hausdorff dimensions of the level sets $E_{φ(n),α}:=\{x\in(0,1): \lim_{n\rightarrow\infty}\frac{s_n(x)}{φ(n)}=α\}$ for $α\geq 0$ and a non-decreasing sequence $\{φ(n)\}_{n=1}^\infty$ have been studied by E. Cesaratto, B. Vallée, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams \emph{et al}. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on $φ(n)$ under which one can expect a $1$-dimensiona…
A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this size we show that, under mild assumptions on the size of the modulus $m$ and on the difference $N^{1/2}-\Abs{\cal A}$ (these quantities should not be too large) the elements of ${\cal A}$ are uniformly distributed in the residue classes mod $m$. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lin…
Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<α|A|$ and $α\ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\mathbb{R}^d$. As an application of the case $A=B$ we give a lower bound on $|A^+|+|A^\times|$, where $A^+$ is the set of sums of distinct elements of $A$ and $A^\times$ is …
Let A be a subset of Z / NZ, and let R be the set of large Fourier coefficients of A. Properties of R have been studied in works of M.-C. Chang and B. Green. Our result is the following : the number of quadruples (r_1, r_2, r_3, r_4) \in R^4 such that r_1 + r_2 = r_3 + r_4 is at least |R|^{2+ε}, ε>0. This statement shows that the set R is highly structured. We also discuss some of the generalizations and applications of our result.
We present two sets of 12 integers that have the same sets of 4-sums. The proof of the fact that a set of 12 numbers is uniquely determined by the set of its 4-sums published 50 years ago is wrong, and we demonstrate an incorrect calculation in it.
Since addition is commutative but subtraction is not, the sumset S+S of a finite set S is predisposed to be smaller than the difference set S-S. In this paper, however, we show that each of the three possibilities (|S+S|>|S-S|, |S+S|=|S-S|, |S+S|<|S-S|) occur for a positive proportion of the subsets of {0, 1, ..., n-1}. We also show that the difference |S+S| - |S-S| can take any integer value, and we show that the expected number of omitted differences is asymptotically 6 while the expected number of missing sums is asymptotically 10. Other data and conjectures on the distribution of these qua…
We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A+A+A contain long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional subspaces if A is a subset of a vector space over a finite field, even if A has density of …
A more sums than differences (MSTD) set is a finite subset S of the integers such |S+S| > |S-S|. We show that the probability that a uniform random subset of {0, 1, ..., n} is an MSTD set approaches some limit rho > 4.28 x 10^{-4}. This improves the previous result of Martin and O'Bryant that there is a lower limit of at least 2 x 10^{-7}. Monte Carlo experiments suggest that rho \approx 4.5 \x 10^{-4}. We present a deterministic algorithm that can compute rho up to arbitrary precision. We also describe the structure of a random MSTD subset S of {0, 1, ..., n}. We formalize the intuition that …
The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2πi (x_1 n+\ldots +x_d n^{d})) \right| \ge N^α $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.
Integer Sets of Large Harmonic Sum Which Avoid Long Arithmetic Progressions
本研究给出数字受限整数集不含k项算术级数的条件，通过大规模搜索找到不含4项与10项算术级数且调和和优于传统贪心构造的整数集，可高效计算相关集合与调和和。
\medskip\noindent\textbf{Résumé.} Soit $l$ un entier et $\ors=(s_1, \dots, s_l)$ une séquence d'entiers positifs. Dans ce document, nous étudierons les propriétés arithmétique de sommes harmoniques multiples $H(\ors; n)$, qui est le $n$-ème somme partielle de la valeur de la série multiple zeta $ζ(\ors)$. On conjecture que pour tout $\ors$ et de tous les premiers $p$, il n'y a que de nombreux finitely $p$-partie intégrante sommes $H(\ors,n)$. Ceci généralise une conjecture de Eswarathasan et Levine et Boyd pour la série harmonique. Nous fournissons beaucoup d'éléments de preuve pour cette conj…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。