arXiv subjects

Jun-Yu Zhou

Publications and source records attributed to Jun-Yu Zhou.

3 recordsLinked to original sources

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.

math.NT↗

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 $D=B_{l-1} \cup\left(2^{l-1}+1+A_{l-1}\right)$.

math.NT↗

arXiv2020

On the structure of the $h$-fold sumsets

Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \geq 2$, and~$A=\left\{a_{0}, a_{1}, \ldots, a_{k}\right\}$ is a finite set of integers such that~$0=a_{0}<a_{1}<\cdots<a_{k}$ and $\gcd\left(a_{1}, a_2,\ldots, a_{k}\right)=1,$ then there exist integers ~$c_{t},d_{t}$ and sets~$C_{t}\subseteq[0, c_{t}-2]$, $D_{t} \subseteq[0, d_{t}-2]$ such that $$(h A)^{(t)}=C_{t} \cup\left[c_{t}, h a_{k}-d_{t}\right] \cup\left(h a_{k-1}-D_{t}\right) $$ for all~$h \geq\sum_{i=2}^{k}(ta_{i}-1)-1.$ This improves a recent result of Nathanson with the bound $h \geq (k-1)\left(t a_{k}-1\right) a_{k}+1$.

math.NT↗