Let N=0,1,2,…. For an integer k>1, a set A⊆ N, and n∈ N, define R_1,k(A,n)=#(a_1,a_2)∈ A^2:a_1+ka_2=n. For integers k>1 and t≥ 1, let f_k(t) be the number of sets A⊆ N such that R_1,k(A,n)=R_1,k( N setminus A,n) for every integer n≥ t. Determine whether, for every pair of integers k,l>1, there exists an integer t_0 such that f_k(t)=f_l(t) for all integers t≥ t_0.

Yang–Chen problem on weighted representation functions

Let $\mathbb N=\{0,1,2,\ldots\}$ . For an integer $k>1$ , a set $A\subseteq\mathbb N$ , and $n\in\mathbb N$ , define $R_{1,k}(A,n)=\#\{(a_1,a_2)\in A^2:a_1+ka_2=n\}$ . For integers $k>1$ and $t\ge 1$ , let $f_k(t)$ be the number of sets $A\subseteq\mathbb N$ such that $R_{1,k}(A,n)=R_{1,k}(\mathbb N\setminus A,n)$ for every integer $n\ge t$ . Determine whether, for every pair of integers $k,l>1$ , there exists an integer $t_0$ such that $f_k(t)=f_l(t)$ for all integers $t\ge t_0$ .

References

Primary source

arXiv

Additional references

A problem of Yang and Chen on weighted representation functions — arXiv — Shuang-Shuang Li, Ya-Ting Xu, Xiao-Hui Yan

additive-number-theory combinatorics

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to settle the question negatively by showing that the number of solutions grows at different rates for different weights, but the result has not been independently verified.

Yang and Chen posed related weighted representation-function questions in 2012. The newer question asks whether the counting functions $f_k(t)$ and $f_l(t)$ eventually agree for all integers $k,l>1$ .

Known results

Qu, 2016: the representation function tends to infinity under the eventual-equality hypothesis.
Shi-Qiang Chen: established a logarithmic lower bound, $\liminf_{n\to\infty}r_{1,k}(A,n)/\log n>0$ .
Chen, Ding, Lü, and Zhang, 2024: proved the stronger bound $\liminf_{n\to\infty}r_{1,k}(A,n)/n>0$ for every $k\ge 2$ , with the correct linear order.

September 17, 2026 preprint

Li, Xu, and Yan claim that $f_k(t)\asymp_k 2^t/t^{k/2}$ . This gives a negative answer to the eventual-equality question for different weights and is currently an unrefereed preprint.

Current status (as of September 2026): The claimed asymptotic formula would settle the $f_k(t)$ question negatively, but it remains unverified; earlier weighted representation-function questions have established affirmative and linear-growth results.

Sources

No solutions have been posted yet.