Javier Cilleruelo
We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solubility of some equations and distribution of sequences in small intervals. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.
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We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Koulantzakis' idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo & Ruzsa & Trujillo, Jia, and Habsieger & Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.
In this paper we prove the existence of Sidon sets which are asymptotic bases of order 4 by using probabilistic methods.
We study the maximum size of Sidon sets in unions of integers intervals. If $A\subseteq\mathbb{N}$ is the union of two intervals and if $\left| A \right|=n$ (where $\left| A \right|$ denotes the cardinality of $A$), we prove that $A$ contains a Sidon set of size at least $0, 876\sqrt{n}$. On the other hand, by using the small differences technique, we establish a bound of the maximum size of Sidon sets in the union of $k$ intervals.
Ramsey-type problems for generalised Sidon sets
受Erdős等学者相关问题启发，研究者证明了具备额外Ramsey型性质的广义Sidon集是存在的，推进了相关组合数论问题的研究。
We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.
Generalising results of Erdős-Freud and Lindström, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution.
We prove that if $A$ is an infinite multiplicative Sidon set, then $\liminf\limits_{n\to \infty}\frac{|A(n)|-π(n)}{\frac{n^{3/4}}{(\log n)^3}}<\infty$ and construct an infinite multiplicative Sidon set satisfying $\liminf\limits_{n\to \infty}\frac{|A(n)|-π(n)}{\frac{n^{3/4}}{(\log n)^3}}>0$.
We give a construction of a set $A \subset \mathbb N$ such that any subset $A' \subset A$ with $|A'| \gg |A|^{2/3}$ is neither an additive nor multiplicative Sidon set. In doing so, we refute a conjecture of Klurman and Pohoata.
We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.
Let X be a subset of an abelian group and a_1,...,a_h,a'_1,...,a'_h a sequence of 2h elements of X such that a_1 + ... + a_h = a'_1 + ... + a'_h. The set X is a Sidon set of order h if, after renumbering, a_i = a'_i for i = 1,..., h. For k \leq h, the set X is a generalized Sidon set of order (h,k), if, after renumbering, a_i = a'_i for i = 1,..., k. It is proved that if X is a generalized Sidon set of order (2h-1,h-1), then the maximal Sidon sets of order h contained in X have the same cardinality. Moreover, X is a matroid where the independent subsets of X are the Sidon sets of order h.
Let $φ(x_1,\ldots, x_h) = c_1 x_1 + \cdots + c_h x_h $ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $φ$-Sidon set if, for all $h$-tuples $(a_1,\ldots, a_h) \in A^h$ and $ (a'_1,\ldots, a'_h) \in A^h$, the relation $φ(a_1,\ldots, a_h) = φ(a'_1,\ldots, a'_h)$ implies $(a_1,\ldots, a_h) = (a'_1,\ldots, a'_h)$. There exist infinite Sidon sets for the linear form $φ$ if and only if the set of coefficients of $φ$ has distinct subset sums. In a normed vector space with $φ$-Sidon sets, every infinite sequ…
Determining Sidon Polynomials on Sidon Sets over $\mathbb{F}_q\times \mathbb{F}_q$
本文针对有限域直积上的西顿集合问题，提出多项式判定准则，证明某些单项式与三次多项式无法生成西顿集合，并揭示其与平面多项式的联系。
The apparent structure of dense Sidon sets
本文揭示稠密 Sidon 集与射影平面之间的构造性联系，分类了来自 Desargues 平面的情形，并猜想所有稠密 Sidon 集均由此产生，同时列举了多种代数来源的较小 Sidon 集。
Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.
A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach, we prove that there exists a perfect difference set A such that A(x) >> x^{\sqrt{2}-1-o(1)}. We also prove that there exists a perfect difference set A such that limsup_{x\to \infty}A(x)/\sqrt x\geq 1/\sqrt 2.
We show that for any finite set $A$ and an arbitrary $\varepsilon>0$ there is $k=k(\varepsilon)$ such that the higher energy ${\mathsf{E}}_k(A)$ is at most $|A|^{k+\varepsilon}$ unless $A$ has a very specific structure. As an application we obtain that any finite subset $A$ of the real numbers or the prime field either contains an additive Sidon--type subset of size $|A|^{1/2+c}$ or a multiplicative Sidon--type subset of size $|A|^{1/2+c}$.
For positive integers $d$ and $n$, let $[n]^d$ be the set of all vectors $(a_1,a_2,\dots, a_d)$, where $a_i$ is an integer with $0\leq a_i\leq n-1$. A subset $S$ of $[n]^d$ is called a \emph{Sidon set} if all sums of two (not necessarily distinct) vectors in $S$ are distinct. In this paper, we estimate two numbers related to the maximum size of Sidon sets in $[n]^d$. First, let $\mathcal{Z}_{n,d}$ be the number of all Sidon sets in $[n]^d$. We show that $\log (\mathcal{Z}_{n,d})=Θ(n^{d/2})$, where the constants of $Θ$ depend only on $d$. Next, we estimate the maximum size of Sidon sets contain…
A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erdős in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the lower order term: $π(n)+Θ(\frac{n^{3/4}}{(\log n)^{3/2}})$. We show that the number of multiplicative Sidon subsets of $\{1,\ldots, n\}$ is $T(n)\cdot 2^{Θ(\frac{n^{3/4}}{(\log n)^{3/2}})}$ for a certain function $T(n)\approx 2^{1.815π(n)}$ which we specify. This is a rare example in which the order of magnitude of the lower order term in the exponent is determine…
A set of integers $S \subset \mathbb{N}$ is an $α$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $α$, more specifically if $| (x+w) - (y+z) | \geq \max \{ x^α,y^α,z^α,w^α\}$ for every $x,y,z,w \in S$ satisfying $\max \{x,w\} \neq \max \{y,z\}$. We obtain a new lower bound for the growth of $α$-strong infinite Sidon sets when $0 \leq α< 1$. We also further extend that notion in a natural way by obtaining the first non-trivial bound for $α$-strong infinite $B_h$ sets. In both cases, we study the implications of these bounds for the density …
Existence of a Sidon set for the distinct distance constant
研究证明了存在一个Sidon集，其倒数和等于distinct distance常数，并改进了该常数的最佳界。
We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots,n\}$ is at most $π(n)+π(n/2)+n^{2/3}(\log n )^{2^{1/3}-1/3+o(1)}$, which improves the previously known best bound $π(n)+π(n/2)+cn^{2/3}\log n/\log\log n$.
We report new examples of Sidon sets in abelian groups arising from algebraic geometry.
For a subset $A \subseteq [N]$, we define the representation function $ r_{A-A}(d) := \#\{(a,a') \in A \times A : d = a - a'\}$ and define $M_D(A) := \max_{1 \leq d < D} r_{A-A}(d)$ for $D>1$. We study the smallest possible value of $M_D(A)$ as $A$ ranges over all possible subsets of $[N]$ with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of $D$. We shall also see how this problem connects to a well-known problem about generalized Sidon sets.
Gaps in Multiplicative Sidon Sets II
本文改进了乘法Sidon集最大间隙的上界，将指数从约0.47降至10/33≈0.303。
Here is a direct problem for Sidon sets: Given a linear form $φ= c_1 x_1 + \cdots + c_h x_h $, construct and describe sets $A$ that are Sidon sets for $φ$. This paper considers an inverse problem for Sidon sets: Given a set $A$, determine the linear forms $φ$ such that $A$ is a Sidon set for $φ$.
On the size of finite Sidon sets
本文改进有限Sidon集直径的下界，得到更紧的数值估计，并采用逻辑上更简单的方法，优于现有结果。
Sidon sets in algebraic geometry
本文研究了代数几何中的 Sidon 集，通过引入曲线的广义 jacobians 构造出 abelian 群中的新实例，并分析了其在大小与结构方面的性质。
A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1,2,...,n} are B_2[g]-sets, in the sense that if B_2[g](k,n) denotes the number of B_2[g]-sets of cardinality k contained in the interval {1,2,...,n}, then lim_{n\to\infty} B_2[g](k,n)/\binom{n}{k} = 1 if k = o(n^{g/(2g+2)}).
In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.
Sidon sets with $Δ$-separated sumsets in additive number theory
该研究针对整数区间内最大Δ-分离西顿集的基数，推导了其上下界，属于加法数论中西顿集相关的理论研究。
Is it true that most sets are Sidon?
本文否定地回答了“大多数集合是否为 Sidon 集”这一问题。
A set $S\subset\{1,2,...,n\}$ is called a Sidon set if all the sums $a+b~~(a,b\in S)$ are different. Let $S_n$ be the largest cardinality of the Sidon sets in $\{1,2,...,n\}$. In a former article, the author proved the following asymptotic formula $$\sum_{a\in S,~|S|=S_n}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}),$$ where $\varepsilon>0$ is an arbitrary small constant. In this note, we give an extension of the above formula. We show that $$\sum_{a\in S,~|S|=S_n}a^{\ell}=\frac{1}{\ell+1}n^{\ell+1/2}+O\left(n^{\ell+61/160}\right)$$ for any positive integers $\ell$. Besides, we also consider …
Fix integers $b>a\geq1$ with $g:=\gcd(a,b)$. A set $S\subseteq\mathbb{N}$ is \emph{$\{a,b\}$-multiplicative} if $ax\neq by$ for all $x,y\in S$. For all $n$, we determine an $\{a,b\}$-multiplicative set with maximum cardinality in $[n]$, and conclude that the maximum density of an $\{a,b\}$-multiplicative set is $\frac{b}{b+g}$. For $A, B \subseteq \mathbb{N}$, a set $S\subseteq\mathbb{N}$ is \emph{$\{A,B\}$-multiplicative} if $ax=by$ implies $a = b$ and $x = y$ for all $a\in A$ and $b\in B$, and $x,y\in S$. For $1 < a < b < c$ and $a, b, c$ coprime, we give an O(1) time algorithm to approximat…
Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof
该研究发现了一个反例，推翻了埃德尔斯关于有限西顿集可扩展为完美差集的猜想，并通过人机协作验证了反例的正确性。
Let $f$ be a nonnegative function supported on $(-1/4, 1/4)$. We show $$ \sup_{x \in \mathbb{R}}{\int_{\mathbb{R}}{f(t)f(x-t)dt}} \geq 1.28\left(\int_{-1/4}^{1/4}{f(x)dx} \right)^2,$$ where 1.28 improves on a series of earlier results. The inequality arises naturally in additive combinatorics in the study of Sidon sets. We derive a relaxation of the problem that reduces to a finite number of cases and yields slightly stronger results. Our approach should be able to prove lower bounds that are arbitrary close to the sharp result. Currently, the bottleneck in our approach is runtime: new ideas m…
Let $F$ be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph $G$ with no bichromatic subgraph in $F$ is $\F$-free. The $F$-free chromatic number $χ(G,F)$ of a graph $G$ is the minimum number of colours in an $F$-free colouring of $G$. For appropriate choices of $F$, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies $F$-free colourings of the cartesian product of graphs. Let $H$ be the cartesian product of the graphs $G_1,G_2,..…
On the Thickness of Infinite Generalized Sidon Sets, II
研究关于无限广义西顿集厚度问题，针对偶数\(h\)及\(B_h\)集，通过特定证明得出\(\liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt[h]{n/\log n}}\)的不等式，给出其上限。
For $h \ge 2$ and an infinite set of positive integers $A$, let $R_{A,h}(n)$ denote the number of solutions of the equation $a_{1} + a_{2} + \dots{} + a_{h} = n, a_{1} \in A, \dots{} ,a_{h} \in A, a_{1} < a_{2} < \dots{} < a_{h}.$ In this paper we prove the existence of a set $A$ formed by perfect powers with almost possible maximal density such that $R_{A,h}(n)$ is bounded by using probabilistic methods.
A combinatorial large sieve for Sidon sets, distances, and norm forms
提出一种针对有界代数重数集合的组合大筛法，通过模小素数分裂产生碰撞与全局有界性矛盾，证明Sidon平方集、无重复距离网格点集等问题的超多对数上界。
On the Thickness of Infinite Generalized Sidon Sets, I
研究无限Sidon集（γ-Golomb尺）的渐近密度，证明上界为2√(γ/log2)√(n/logn)，并构造下界为(1/√2)√γ的下界。
A subset $A$ of an additive abelian group is an $h$-Sidon set if every element in the $h$-fold sumset $hA$ has a unique representation as the sum of $h$ not necessarily distinct elements of $A$. Let $\mathbf{F}$ be a field of characteristic 0 with a nontrivial absolute value, and let $A = \{a_i :i \in \mathbf{N} \}$ and $B = \{b_i :i \in \mathbf{N} \}$ be subsets of $\mathbf{F}$. Let $\varepsilon = \{ \varepsilon_i:i \in \mathbf{N} \}$, where $\varepsilon_i > 0$ for all $i \in \mathbf{N}$. The set $B$ is an $\varepsilon$-perturbation of $A$ if $|b_i-a_i| < \varepsilon_i$ for all $i \in \mathbf…
Two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) are described, the density of which exceeds that of any prior construction by a factor of at least $ 2^{\frac{n}{\ln n}(1 + o(1))} $ when $ n \to \infty $. The first family of lattices is explicit and is obtained by applying Construction A to a class of Reed-Solomon codes. The second family has subexponential construction complexity and is based on the notion of Sidon sets in finite Abelian groups. The construction based on Sidon sets also gives the highest known asymptotic density of packing discrete…
Dense finite Sidon sets on arithmetic progressions
本文针对稠密有限Sidon集合，利用Ortega和Prendiville的极值Sidon集傅里叶均匀性，证明了其元素幂和与秩加权和的渐近公式，去除了先前结果中的额外假设。
In the ElGamal signature and encryption schemes, an element $x$ of the underlying group $G = \mathbb{Z}_p^\times = \{1, \ldots, p-1 \}$ for a prime $p$ is also considered as an exponent, for example in $g^x$, where $g$ is a generator of G. This ElGamal map $x \mapsto g^x$ is poorly understood, and one may wonder whether it has some randomness properties. The underlying map from $G$ to $\mathbb{Z}_{p-1}$ with $x \mapsto x$ is trivial from a computer science point of view, but does not seem to have any mathematical structure. This work presents two pieces of evidence for randomness. Firstly, exp…
Let $h \geq 2$ and let ${ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of sets of integers. For nonzero integers $c_1,\ldots, c_h$, consider the linear form $φ= c_1 x_1 + c_2x_2 + \cdots + c_h x_h$. The \emph{representation function} $R_{ \mathcal{A},φ}(n)$ counts the number of $h$-tuples $(a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h$ such that $φ(a_1,\ldots, a_h) = n$. The $h$-tuple $\mathcal{A}$ is a \emph{$φ$-Sidon system of multiplicity $g$} if $R_{\mathcal A,φ}(n) \leq g$ for all $n \in \mathbf{Z}$. For every positive integer $g$, let $F_{φ,g}(n)$ denote the largest integer $q$ …
A symmetric subset of the reals is one that remains invariant under some reflection x --> c-x. Given 0 < x < 1, there exists a real number D(x) with the following property: if 0 < d < D(x), then every subset of [0,1] with measure x contains a symmetric subset with measure d, while if d > D(x), then there exists a subset of [0,1] with measure x that does not contain a symmetric subset with measure d. In this paper we establish upper and lower bounds for D(x) of the same order of magnitude: for example, we prove that D(x) = 2x - 1 for 11/16 < x < 1 and that 0.59 x^2 < D(x) < 0.8 x^2 for 0 < x < …
Linear equations and chromatic thresholds in $B_h$ sets
研究在近最大尺寸的 $B_h$ 集合中，禁止特定线性方程解的存在性对集合大小的影响，并证明极值 $B_h$ 集合的傅里叶伪随机性。
On the Sidon tails of $\left\{\lfloor x^n\rfloor\right\}$
本文证明了对于几乎所有x∈(1,2)，{⌊xⁿ⌋}的尾部是Sidon集合，并展示了存在某些x和r使得对应的集合不具有Sidon尾部。
A $ B_h $ set (or Sidon set of order $ h $) in an Abelian group $ G $ is any subset $ \{b_0, b_1, \ldots,b_{n}\} $ of $ G $ with the property that all the sums $ b_{i_1} + \cdots + b_{i_h} $ are different up to the order of the summands. Let $ ϕ(h,n) $ denote the order of the smallest Abelian group containing a $ B_h $ set of cardinality $ n + 1 $. It is shown that \[ \lim_{h \to \infty} \frac{ ϕ(h,n) }{ h^n } = \frac{1}{n! δ_L(\triangle^n)} , \] where $ δ_L(\triangle^n) $ is the lattice packing density of an $ n $-simplex in Euclidean space. This determines the asymptotics exactly in cases wh…
Infinite Sidon-type sets for zero-sum linear forms
研究零和线性形式下集合\(A\)的密度限制，通过不同条件得出表示数与集合密度的关系，如特定\(\mathbf{b}\)形式及一般\(\mathbf{b}\)在间隙条件下的相关结论，恢复了关于\(B_{2k}\)序列的定理。
Finding large additive and multiplicative Sidon sets in sets of integers
本文研究整数有限集合中大型加性与乘性 $B_h[g]$ 集的存在性，证明二者最大规模至少为 $|A|^{(1+δ)/h}$，并给出 $h=2$ 时 $g=31$ 及大 $h$ 时 $g=1$ 的具体常数，推进了相关猜想。
Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 [g] sets and defined by the fact that there are at most g ways (up to reordering the summands) to represent a given integer as a sum of two elements of the set, are much more difficult to handle and not as well understood. In this article, using a numerical approach, we improve the best upper estimates …
On Sidon sets with squares, cubes and quartics in short intervals
本文研究了短区间内含平方、立方和四次方的Sidon集，证明了相关方程在特定范围内的无解性和存在性。
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。
In 2010, Cilleruelo, Ruzsa, and Vinuesa established a surprising connection between the maximum possible size of a generalized Sidon set in the first $N$ natural numbers and the optimal constant in an ``analogous'' problem concerning nonnegative-valued functions on $[0,1]$ with autoconvolution integral uniformly bounded above. Answering a recent question of Barnard and Steinerberger, we prove the corresponding dual result about the minimum size of a so-called generalized difference set that covers the first $N$ natural numbers and the optimal constant in an analogous problem concerning nonnega…
In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and we apply our result to give an easy proof of a result which says that dense subsets A and B of {1,2,...,x} always produce asymptotically the expected number of x^r - smooth sums a+b, where a in A and b in B. Recall that a number n is said to be y-smooth if all its prime divisors are at most y.
We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán's theorem, Szemerédi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Turán's theorem to the random setting by showing that for every $ε> 0$ and every positive integer $t \geq 3$ there exists a constant $C$ such that, if $G$ is a random graph on $n$ vertices where each edge is chosen independently with probability at least $C n^{-2/(t+1)}$, then, with probability tending to $1$ as $n$ tends to infinity, every subgra…
In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely $\mathrm{Nil}_d{-}\mathrm{Bohr}$ and $\mathrm{SG}_k$, as introduced by Host and Kra. We prove that any $\mathrm{Nil}_d{-}\mathrm{Bohr}_0$ set is necessarily $\mathrm{SG}_k$ where ${k}$ is effectively bounded in terms of $d$. This partially resolves a conjecture of Host and Kra.
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