arXiv subjects

Csaba Sándor

Publications and source records attributed to Csaba Sándor.

At least 19 recordsLinked to original sources

arXiv2026

Cross representations of additive complements of $r$-th powers

Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$ f_r(n)=\#\big\{(w,m^r)\in \mathcal{W}_r\times \mathcal{S}_r: n=w+m^r\big\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \sum_{n\le N}f_r(n)-N\gg_r N^{1-\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \sum_{n\le N}f_2(n)-N\gg N^{3/4-o(1)}, $$ which improves the previous bound $N^{1/2}$ due to Ding, Sun, Wang and Xia.

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arXiv2026

On Sárközy-Sós Theorem related to representation functions

Let $\mathbb{N}_0$ be the set of all nonnegative integers. For a nonempty set $\mathcal{A}\subseteq \mathbb{N}_0$ and integers $n,h\ge 2$, let $r_{h}(\mathcal{A},n)$ be the number of representations of $n$ as $a_1+\cdots+a_h$, where $a_1\le \cdots\le a_h$ and $a_i\in \mathcal{A}$ for $i=1,\cdots,h$. In 2016, Chen and Tang showed that, for any given distinct positive integers $u_1,\cdots,u_k$ and positive rational numbers $α_1,\cdots,α_k$ with $α_1+\cdots+α_k=1$, there are infinitely many sets $\mathcal{A}\subseteq \mathbb{N}_0$ such that $r_{h}(\mathcal{A},n)\ge 1$ for all nonnegative integers $n$ and the set of $n$ with $r_{h}(\mathcal{A},n)=u_i$ has density $α_i$ for all integer $i=1,\cdots,k$. In this paper, we consider the irrational numbers $α_i$ as well. As a main result, we prove that, for any nonnegative numbers $α_0,\cdots,α_m$ with $α_0+\cdots+α_m=1$, there are infinitely many sets $\mathcal{A}\subseteq \mathbb{N}_0$ such that the set of $n$ with $r_{2}(\mathcal{A},n)=i$ has density $α_i$ for all integer $i=0,\cdots,m$. Other related results are also contained.

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arXiv2026

On the Monotonicity of Higher-Fold Representation Functions

For a positive integer $h$, let $R_{A,h}(n)$ denote the number of ordered representations $n=s_1+\cdots+s_h$ with all $s_i\in A$. Let \[ B=\{0\}\cup\{m\ge 1:\text{ the base-4 expansion of }m\text{ begins with }1\text{ or }2\}. \] Shallit proved that $R_{B,3}(n)$ is strictly increasing, thereby disproving a 2002 conjecture of Dombi. In this paper, by using linear bounds for $R_{B,3}(n+1)-R_{B,3}(n)$ and a convolution argument, we prove the polynomial order of $R_{B,h}(n+1)-R_{B,h}(n)$ for every integer $h\ge 3$. More precisely, for every integer $h\ge 3$, there exist constants $c_h,C_h>0$, depending only on $h$, such that \[ c_h n^{h-2}\le R_{B,h}(n+1)-R_{B,h}(n)\le C_h n^{h-2} \] for all integers $n\ge 1$. We also construct a co-infinite set $C\subset\mathbb N$ satisfying $\lim_{n\to\infty}C(n)/n=1$ such that $R_{C,h}(n)$ is strictly increasing for every integer $h\ge 3$. This answers a problem of Dombi posed in 2002. We also pose some problems for further research.

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arXiv2026

On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity

We consider the equality of the values of the $n$th and $k$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. For $k < n$, we prove that this equation always has a solution, but only finitely many solutions. Furthermore, we consider the equality of the values of the $n$th and $(n-2)$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. In particular, we show that the number of solutions of this equation tends to infinity if $n$ tends to infinity.

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arXiv2026

Product representations of perfect powers

Let $ρ_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right) +O_d(π(N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right)$. This answers a question of Verstraëte.

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arXiv2025

Identical representation functions of linear forms

For a set of natural numbers $A$, let $R_{A}(n)$ be the number of representations of a natural number $n$ as the sum of two terms from $A$. Many years ago, Nathanson studied the conditions for the set $A$ and $B$ of natural numbers that are needed to guarantee that $R_{A}(n) = R_{B}(n)$ for every positive integer $n$. In the last decades, similar questions have been studied by many authors. In this paper, we extend Nathanson's result to representation functions associated to linear forms and we study related problems.

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arXiv2024

Equal Sum and Product Problem III

Denote by $N(n)$ the number of integer solutions $(x_1,\,x_2,\ldots ,x_n)$ of the equation $x_1+x_2+\ldots+x_n=x_1x_2\cdot\ldots\cdot x_n$ such that $x_1\ge x_2\ge\ldots\ge x_n\ge 1$, $n \in \mathbb{Z}^+$. The aim of this paper are is twofold: first we present an asymptotic formula for $\sum\limits_{2\le n\le x}N(n)$, then we verify that the counting function $N(n)$ takes very large value compared to its average value.

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arXiv2024

On the largest value of the solutions of Erdős's last equation

Let $n$ be a positive integer. The Diophantine equation $n(x_1+x_2+\dots +x_n)=x_1x_2\dots x_n$, $1 \le x_1\le x_2\le \dots \le x_n$ is called Erdős's last equation. We prove that $x_n\to \infty $ as $n\to \infty$ and determine all tuples $(n,x_1,\dots ,x_n)$ with $x_n\le 10$.

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arXiv2024

Product representation of perfect cubes

Let $F_{k,d}(n)$ be the maximal size of a set ${A}\subseteq [n]$ such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] has no solution with $a_1,a_2,\ldots,a_k\in {A}$ and integer $x$. Erdős, Sárközy and T. Sós studied $F_{k,2}$, and gave bounds when $k=2,3,4,6$ and also in the general case. We study the problem for $d=3$, and provide bounds for $k=2,3,4,6$ and $9$, furthermore, in the general case, as well. In particular, we refute an 18 years old conjecture of Verstraëte. We also introduce another function $f_{k,d}$ closely related to $F_{k,d}$: While the original problem requires $a_1, \ldots , a_k$ to all be distinct, we can relax this and only require that the multiset of the $a_i$'s cannot be partitioned into $d$-tuples where each $d$-tuple consists of $d$ copies of the same number.

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arXiv2023

Representation functions in the set of natural numbers

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 $n=s+s'$, $s, s'\in S$, $s<s'$. In this paper, we determine the structure of all sets $A$ and $B$ such that $A\cup B=\mathbb{N}\setminus\{r+mk:k\in\mathbb{N}\}$, $A\cap B=\emptyset$ and $R_{A}(n)=R_{B}(n)$ for every positive integer $n$, where $m$ and $r$ are two integers with $m\ge 2$ and $r\ge 0$.

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arXiv2023

Lagrange-like spectrum of perfect additive complements

Two infinite sets $A$ and $B$ of non-negative integers are called \emph{perfect additive complements of non-negative integers}, if every non-negative integer can be uniquely expressed as the sum of elements from $A$ and $B$. In this paper, we define a Lagrange-like spectrum of the perfect additive complements ($\mathfrak{L} $ for short). As a main result, we obtain the smallest accumulation point of the set $\mathfrak{L} $ and prove that the set $\mathfrak{L} $ is closed. Other related results and problems are also contained.

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arXiv2023

Multiplicative complements II

In this paper we prove that if $A$ and $B$ are infinite subsets of positive integers such that every positive integer $n$ can be written as $n=ab$, $a\in A$, $b\in B$, then $\displaystyle \lim_{x\to \infty}\frac{A(x)B(x)}{x}=\infty $. We also prove many other results about sets like this.

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arXiv2023

Unique representations of integers by linear forms

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underlineλ = (λ_{1}, \dots{} ,λ_{k})$ with $1 \le λ_{1} < λ_{2} < \dots{} < λ_{k}$, we define the additive representation function $R_{A,\underlineλ}(n) = |\{(a_{1}, \dots{} ,a_{k})\in A^{k}: λ_{1}a_{1} + \dots{} + λ_{k}a_{k} = n\}|$. For $k = 2$, Moser constructed a set $A$ of nonnegative integers such that $R_{A,\underlineλ}(n) = 1$ holds for every nonnegative integer $n$. In this paper we characterize all the $k$-tuples $\underlineλ$ and the sets $A$ of nonnegative integers with $R_{A,\underlineλ}(n) = 1$ for every integer $n\ge 0$.

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arXiv2023

On monotone increasing representation functions

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$ denote the counting function of $A$.Bell and Shallit recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{\frac{k-2}{k}-ε})$ for some $ε>0$, then $R_{\mathbb{N}\setminus A,k}(n)$ is eventually strictly increasing. In this paper, we improve this result to $A(n)=O(n^{\frac{k-2}{k-1}})$. We also give an example to show that this bound is best possible.

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arXiv2023

On a problem of Nathanson related to minimal asymptotic bases of order $h$

For integer $h\geq2$ and $A\subseteq\mathbb{N}$, we define $hA$ to be all integers which can be written as a sum of $h$ elements of $A$. The set $A$ is called an asymptotic basis of order $h$ if $n\in hA$ for all sufficiently large integers $n$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. For $W\subseteq\mathbb{N}$, denote by $\mathcal{F}^*(W)$ the set of all finite, nonempty subsets of $W$. Let $A(W)$ be the set of all numbers of the form $\sum_{f \in F} 2^f$, where $F \in \mathcal{F}^*(W)$. In this paper, we give some characterizations of the partitions $\mathbb{N}=W_1\cup\cdots \cup W_h$ with the property that $A=A(W_1)\cup\cdots \cup A(W_{h})$ is a minimal asymptotic basis of order $h$. This generalizes a result of Chen and Chen, recent result of Ling and Tang, and also recent result of Sun.

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arXiv2023

Polynomial Schur's theorem

We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. For polynomials that are not $2$-Ramsey, we characterise all $2$-colourings of $\mathbb{N}$ that are not $2$-Ramsey, revealing that certain divisibility barrier is the only obstruction to $2$-Ramseyness for $x+y=p(z)$.

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arXiv2022

On function $SX$ of additive complements

Two sets $A,B$ of nonnegative integers are called \emph{additive complements}, if all sufficiently large integers can be expressed as the sum of two elements from $A$ and $B$. We further call $A,B$ \emph{perfect additive complements} if every nonnegative integer can be uniquely expressed as the sum of two elements from $A$ and $B$. Let $A(x)$ be the counting function of $A$. In this paper, we focus on the function $SX$, where $SX=\limsup_{x\rightarrow\infty}\frac{\max\{A(x),B(x)\}}{\sqrt{x}}$ was introduced by Erdős and Freud in 1984. As a main result, we determine the value of $SX$ for perfect additive complements and further fix the infimum. We also give the absolute lower bound of $SX$ for additive complements.

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arXiv2022

Additive completition of thin sets

Two sets $A,B$ of positive integers are called \emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow1$. Let $A=\{a_1<a_2<\cdots\}$ be a set of positive integers. Denote $A(x)$ by the counting function of $A$ and $a^*(x)$ by the largest element in $A\bigcap [1,x]$. Following the work of Ruzsa and Chen-Fang, we prove that, for exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$, we have $A(x)B(x)-x\ge \frac{a^*(x)}{A(x)}+o\left(\frac{a^*(x)}{A(x)^2}\right)$ as $x\rightarrow +\infty$. On the other hand, we also construct exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$ such that $A(x)B(x)-x\le \frac{a^*(x)}{A(x)}+(1+o(1))\left(\frac{a^*(x)}{A(x)^2}\right)$ holds for infinitely many positive integers $x$.

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