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Csaba Sándor

Publications and source records attributed to Csaba Sándor.

At least 37 records · Page 2Linked to original sources
arXiv2022

On disjoint sets

Two sets of nonnegative integers $A=\{a_1 \varepsilon\sqrt{x}$ and $B(x)>\varepsilon\sqrt{x}$ for some $\varepsilon>0$, which answered a problem posed by Erd\H os and Graham. In this paper, following Erdős and Freud's work, we explore further properties for disjoint sets. As a main result, we prove that, for disjoint sets $A$ and $B$, assume that $\{x_1<x_2<\cdots\}$ is a set of positive integers such that $\frac{A(x_n)B(x_n)}{x_n}\rightarrow 2$ as $x_n\to \infty$, then, (i) for any $0<c_1<c_2<1,$ $c_1x_n\le y\le c_2x_n$, we have $\frac{A(y)B(y)}{y}\rightarrow1$ as $n\rightarrow \infty$; (ii) for any $1<c_3<c_4<2,$ $c_3x_n\le y\le c_4x_n$, we have $A(y)B(y)=(2+o(1))x_n$ as $n\rightarrow \infty$.

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arXiv2022

On sets with sum and difference structure

For nonempty sets $A,B$ of nonnegative integers and an integer $n$, let $r_{A,B}(n)$ be the number of representations of $n$ as $a+b$ and $d_{A,B}(n)$ be the number of representations of $n$ as $a-b$, where $a\in A, b\in B$. In this paper, we determine the sets $A,B$ such that $r_{A,B}(n)=1$ for every nonnegative integer $n$. We also consider the \emph{difference} structure and prove that: there exist sets $A$ and $B$ of nonnegative integers such that $r_{A,B}(n)\ge 1$ for all large $n$, $A(x)B(x)=(1+o(1))x$ and for any given nonnegative integer $c$, we have $d_{A,B}(n)=c$ for infinitely many positive integers $n$. Other related results are also contained.

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arXiv2022

Multiplicative complements I

In this paper, we study how dense a multiplicative basis of order $h$ for $\mathbb{Z}^+\!$ can be, improving on earlier results. Upon introducing the notion of a \textit{multiplicative complement}, we present some tight density bounds.

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arXiv2022

On $B_{h}[1]$-sets which are asymptotic bases of order $2h$

Let $h,k \ge 2$ be integers. A set $A$ of positive integers is called asymptotic basis of order $k$ if every large enough positive integer can be written as the sum of $k$ terms from $A$. A set of positive integers $A$ is said to be a $B_{h}[g]$-set if every positive integer can be written as the sum of $h$ terms from $A$ at most $g$ different ways. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h$ by using probabilistic methods.

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arXiv2021

Note on a problem of Nathanson related to the $φ$-Sidon set

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 $φ\left(a_{1}, \ldots, a_{h}\right)=φ\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right)$ implies $\left(a_{1}, \ldots, a_{h}\right)=$ $\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right)$ for all $h$-tuples $\left(a_{1}, \ldots, a_{h}\right) \in A^{h}$ and $\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right) \in A^{h}$. We call $A$ a polynomial perturbation of $B$ if for some $r>0$ and positive integer $k_0$, $|a_k-b_k|< k^r$ holds for all integers $k \geq k_0$. In this paper, for a given set $B$, we prove that there exists a $φ$-Sidon set $A$ of integers that is a polynomial perturbation of $B$. This gives an affirmative answer to a recent problem of Nathanson. Some other results are also proved.

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arXiv2021

On the integer sets with the same representation functions

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s_1+s_2$, $s_1,s_2\in S$ and $s_1<s_2$. Let $A$ be the set of all nonnegative integers which contain an even number of digits $1$ in their binary representations and $B=\mathbb{N}\setminus A$. Put $A_l=A\cap [0,2^l-1]$ and $B_l=B\cap [0,2^l-1]$. In 2017, Kiss and Sándor proved that, if $C\cup D=[0,m]$, $0\in C$ and $C\cap D=\{r\}$, then $R_C(n)=R_D(n)$ for every positive integer $n$ if and only if there exists an integer $l\ge 1$ such that $r=2^{2l}-1$, $m=2^{2l+1}-2$, $C=A_{2l}\cup (2^{2l}-1+B_{2l})$ and $D=B_{2l}\cup (2^{2l}-1+A_{2l})$. This solved a problem of Chen and Lev. In this paper, we prove that, if $C \cup D=[0, m]\setminus \{r\}$ with $0<r<m$, $C \cap D=\emptyset$ and $0 \in C$, then $R_{C}(n)=R_{D}(n)$ for any nonnegative integer $n$ if and only if there exists an integer $l \geq 2$ such that $m=2^{l}$, $r=2^{l-1}$, $C=A_{l-1} \cup\left(2^{l-1}+1+B_{l-1}\right)$ and $D=B_{l-1} \cup\left(2^{l-1}+1+A_{l-1}\right)$.

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arXiv2021

Dense sumsets of Sidon sequences

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.

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arXiv2020

On a problem of Chen and Fang related to infinite additive complements

Two infinite sets $A$ and $B$ of nonnegative integers are called additive complements if their sumset contains every nonnegative integer. In 1964, Danzer constructed infinite additive complements $A$ and $B$ with $A(x)B(x) = (1 + o(1))x$ as $x \rightarrow \infty$, where $A(x)$ and $B(x)$ denote the counting function of the sets $A$ and $B$, respectively. In this paper we solve a problem of Chen and Fang by extending the construction of Danzer.

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arXiv2020

Additive representation functions and discrete convolutions

For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a'$ with $a$, $a'\in A$. Denote by $χ_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to \infty}b_n<1$. In this paper, we prove some Erd\H os--Fuchs-type theorems about the error terms appearing in approximation formulæ for $R_A(n)=\sum_{k=0}^nχ_A(k)χ_A(n-k)$ and $\sum_{n=0}^NR_A(n)$ having principal terms $\sum_{k=0}^nb_kb_{n-k}$ and $\sum_{n=0}^N\sum_{k=0}^nb_kb_{n-k}$, respectively.

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arXiv2020

Generalized asymptotic Sidon basis

Let $h,k \ge 2$ be integers. 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 $B_{h}[g]$ set if all positive integers can be represented as the sum of $h$ terms from $A$ at most $g$ times. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h+1$ by using probabilistic methods.

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arXiv2018

On minimal additive complements of integers

Let $C,W\subseteq \mathbb{Z}$. If $C+W=\mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. Let $X\subseteq \mathbb{N}$. If there exists a positive integer $T$ such that $x+T\in X$ for all sufficiently large integers $x\in X$, then we call $X$ eventually periodic. In this paper, we study the existence of a minimal complement to $W$ when $W$ is eventually periodic or not. This partially answers a problem of Nathanson.

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arXiv2018

Generalizations of some results about the regularity properties of an additive representation function

Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erdős, A. Sárközy and V. T. Sós proved that if $\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty$ then $|Δ_{1}(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and $Δ_{l}$ denotes the $l$-th difference. Their result was extended to $Δ_{l}(R_{A,2}(n))$ for any fixed $l\ge2$. In this paper we give further generalizations of this problem.

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arXiv2017

On generalized Stanley sequences

Let $\mathbb{N}$ denote the set of all nonnegative integers. Let $k\ge 3$ be an integer and $A_{0} = \{a_{1}, \dots{}, a_{t}\}$ $(a_{1} < \ldots< a_{t})$ be a nonnegative set which does not contain an arithmetic progression of length $k$. We denote $A = \{a_{1}, a_{2}, \dots{}\}$ defined by the following greedy algorithm: if $l \ge t$ and $a_{1}, \dots{}, a_{l}$ have already been defined, then $a_{l+1}$ is the smallest integer $a > a_{l}$ such that $\{a_{1}, \dots{}, a_{l}\} \cup \{a\}$ also does not contain a $k$-term arithmetic progression. This sequence $A$ is called the Stanley sequence of order $k$ generated by $A_{0}$. In this paper, we prove some results about various generalizations of the Stanley sequence.

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arXiv2017

On infinite multiplicative Sidon sets

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 $\liminf\limits_{n\to \infty}\frac{|A(n)|-π(n)}{\frac{n^{3/4}}{(\log n)^3}}>0$.

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arXiv2017

On a conjecture of Erdős about sets without $k$ pairwise coprime integers

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) = \text{max}_{A \in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erdős conjectured that $f(n, k) = |E(n,k)|$ for every $n \ge p_{k}$. Recently Chen and Zhou proved some results about this conjecture. In this paper we solve an open problem of Chen and Zhou and prove several related results about the conjecture.

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arXiv2017

On some properties of representation functions related to the Erdős-Turán conjecture

For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The celebrated Erdős-Turán conjecture says that, if $R_A(n)\ge 1$ for all sufficiently large integers $n$, then the representation function $R_A(n)$ cannot be bounded. For any positive integer $m$, Ruzsa's number $R_m$ is defined to be the least positive integer $r$ such that there exists a set $A\subseteq \mathbb{Z}_m$ with $1\le R_A(n)\le r$ for all $n\in \mathbb{Z}_m$. In 2008, Chen proved that $R_{m}\le 288$ for all positive integers $m$. Recently the authors proved that $R_m\ge 6$ for all integers $m\ge 36$. In this paper, we prove that if $A\subseteq \mathbb{Z}_m$ satisfies $R_A(n)\le 5$ for all $n\in \mathbb{Z}_m$, then $|\{g:g\in \mathbb{Z}_m, R_A(g)=0\}|\ge \frac{1}{4}m-\sqrt{5m}$. This improves a recent result of Li and Chen. We also give upper bounds of $|\{g:g\in \mathbb{Z}_m, R_A(g)=i\}|$ for $i=2,4$.

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arXiv2016

A lower bound of Ruzsa's number related to the Erdős-Turán conjecture

For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The celebrated Erdős-Turán conjecture says that, if $R_A(n)\ge 1$ for all sufficiently large integers $n$, then the representation function $R_A(n)$ cannot be bounded. For any positive integer $m$, Ruzsa's number $R_m$ is defined to be the least positive integer $r$ such that there exists a set $A\subseteq \mathbb{Z}_m$ with $1\le R_A(n)\le r$ for all $n\in \mathbb{Z}_m$. In 2008, Chen proved that $R_{m}\le 288$ for all positive integers $m$. In this paper, we prove that $R_m\ge 6$ for all integers $m\ge 36$. We also determine all values of $R_m$ when $m\le 35$.

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