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Quan-Hui Yang

Publications and source records attributed to Quan-Hui Yang.

At least 19 recordsLinked to original sources
arXiv2026

Density and distribution of quotients associated with Goldring's $w$-function

For primes $p,q,r$ that are not all equal, let $n=pqr$ and define Goldring's $w$-function by $w(n)=P(p+q)P(p+r)P(q+r)$, where $P(a)$ denotes the largest prime factor of $a$. In this paper, we resolve three problems of Chen, Shi and Wu. As $n\to\infty$ through such products, the sets of finite limit points of $P(w(n))/P(n)$ and $w(n)/n$ are $[0,1]$ and $[0,\infty)$, respectively. The same conclusions hold when $n$ is restricted either to $p^2q$ with distinct primes $p,q$ or to squarefree products of three primes. Nevertheless, on each of these three sets, the proportion of integers $n\le X$ satisfying $P(w(n))/P(n)>α$ tends to zero as $X\to\infty$, for every fixed $α>0$. We also determine the complete set of finite joint limit points of the two quotients for $n=p^2q$ with $p\ne q$.

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arXiv2026

On a problem of Ma and Chen concerning minimal additive complements

Let $m\geq1$ be an integer and let $W=(m\mathbb N_0+X)\cup U\cup V\subseteq\mathbb Z$ be bounded below, where $X\ne\varnothing$, $[U]_m\subseteq[X]_m$, $[X]_m\cap[V]_m=\varnothing$, and $V$ contains no infinite arithmetic progression with common difference $m$. Here $[S]_m$ denotes the set of residue classes modulo $m$ represented by $S$. Suppose that $W$ satisfies the $\mathcal C$-condition modulo $m$. For $0\leq r<m$, let $V_r=V\cap(r+m\mathbb Z)$, and put $V'=\bigcup_{|V_r|=\infty}V_r$, where the union is over $0\leq r<m$. We prove that $W$ has a minimal additive complement if $V'$ satisfies the big-hole property modulo $m$. This answers a problem posed by Ma and Chen.

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arXiv2026

On the denominators of signed harmonic sums

Let $\eps=(\eps_i)_{i\ge1}$ be a sequence with $\eps_i\in\{-1,1\}$, and let $v_n(\eps)$ be the positive denominator of $\sum_{i=1}^n\eps_i/i$ in lowest terms. We prove that $|\{1\le n\le x:v_n(\eps)\ne v_{n+1}(\eps)\}|=o(x)$ uniformly in $\eps$. In particular, the set of positive integers $n$ with $v_n(\eps)=v_{n+1}(\eps)$ has asymptotic density one for every sequence of signs. This answers a problem of Wu and Chen, \emph{J. Number Theory} \textbf{200} (2019), 397--406.

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arXiv2026

A converse to the Erdős-Fuchs theorem

We prove that there exists $A\subseteq \mathbb{N}$ such that \[ R_A(N)=\fracπ{4}N+ O\!\left(N^{1/4}\sqrt{\log N}\right), \] where $R_A(N)=\#\{(a,b)\in A^2:a+b\le N\}$. This improves the record $O(N^{1/4}\log N)$ obtained by Ruzsa in 1997.

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arXiv2026

On additive complements in natural numbers

We prove that for additive complements $A$ and $B$, if $$ \limsup_{x\to+\infty}\frac{A(x)B(x)}{x}<\frac43, $$ then $$ A(x)B(x)-x\rightarrow+\infty\qquad(x\to+\infty). $$ This improves the earlier upper bound $3-\sqrt{3}$ due to Fang and Chen.

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arXiv2026

Products of Two Integers Avoiding Perfect Powers

For integers $d\geq 3$, let $F_{2,d}(n)$ be the largest size of a subset of $[n]$ containing no two distinct elements whose product is a perfect $d$-th power, and let $f_{2,d}(n)$ denote the analogous quantity when the two elements need not be distinct. Fleiner, Juhász, Kövér, Pach, and Sándor proved that both complements have order $n^{2/3}$ when $d=3$, and asked for a leading constant. They also asked whether, more generally, $n-F_{k,d}(n)$ and $n-f_{k,d}(n)$ have order $n^{k/d}$ for $1 0$ is given explicitly by an Euler product and a polytope volume. In particular, the extra logarithmic factor gives a negative answer to the second question for every $d\geq4$. For $d=3$ we obtain \[ C_3=\frac{π^2}{4} \prod_p\left(1-\frac3{p^2}+\frac2{p^3}\right), \] which answers the first question. The proof uses an exact decomposition into complementary $d$-free kernel classes, a squarefree sieve in multiplicative boxes, and a two-height polytope calculation.

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arXiv2026

On the size of $h$-fold sumsets

Let $h$ be a positive integer, and let $A$ be a finite set of integers. We derive an exact formula for $|hA|$. Furthermore, let $A=\{0,1,\ldots,s,a,b\}$, $1\leq s<a<b$, and write $b=qa+r$ with $0\leq r<a$. By using generating function, we prove that $|hA|$ equals a definite explicit formula expressed in terms of certain truncated binomial coefficients for all positive integers $h$ if and only if $r=0$ or $qs+r\geq a$. This generalizes a result of Nathanson.

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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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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

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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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 a discriminator for the polynomial $f(x)=x^3+x$

Let $Δ(n)$ denote the smallest positive integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m$. The purpose of this paper is to determine $Δ(n)$ for all positive integers $n$.

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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

On a conjecture of Sun involving powers of three

Given a positive integer $n\ge 2$, let $D(n)$ denote the smallest positive integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m^2$. A conjecture of Z.-W. Sun states that $D(n)=3^k$, where $3^k$ is the least power of $3$ no less than $\sqrt{n}$. The purpose of this paper is to confirm this conjecture.

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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 $h \geq (k-1)\left(t a_{k}-1\right) a_{k}+1$.

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arXiv2019

A note on the lower bound of representation functions

For a set $A$ of nonnegative integers, let $R_2(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$. Let $A_0$ be the Thue-Morse sequence and $B_0=\mathbb{N}\setminus A_0$. Let $A\subset \mathbb{N}$ and $N$ be a positive integer such that $R_2(A,n)=R_2(\mathbb{N}\setminus A,n)$ for all $n\geq 2N-1$. Previously, the first author proved that if $|A\cap A_0|=+\infty$ and $|A\cap B_0|=+\infty$, then $R_2(A,n)\geq \frac{n+3}{56N-52}-1$ for all $n\geq 1$. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.

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