发表机构
HSE University（高等经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文针对固定底数的幂乘积多重集，在密度和相关假设下，证明了具有多个表示的整数数量的下界，并应用于素数及两平方和集合，无需乘法独立性假设。
AI 中文摘要
设$b_1,\dots,b_d\ge2$为固定整数，并设$k$为它们的乘法秩。我们研究表示$n=a+b_1^{u_1}\cdots b_d^{u_d}$，其中$a$属于正整数集合$\mathcal{A}$，$u_1,\dots,u_d$为正整数，并分别计数不同的指数元组。在$\mathcal{A}$的密度和相关假设下，我们获得了具有许多此类表示的整数数量的下界，该下界以$d$和$k$表示。特别地，当$\mathcal{A}$为素数集合或可表示为两个平方和的正整数集合时，对于某个$c_1>0$及所有足够大的$x$，正整数$n\le x$中分别有正比例的数至少具有$c_1(\log x)^{d-1}$或$c_1(\log x)^{d-1/2}$个表示。不需要对底数作乘法独立性或互素性假设。
英文摘要
Let $b_1,\dots,b_d\ge2$ be fixed integers, and let $k$ be their multiplicative rank. We study representations $n=a+b_1^{u_1}\cdots b_d^{u_d}$, where $a$ belongs to a set $\mathcal{A}$ of positive integers and $u_1,\dots,u_d$ are positive integers, counting distinct tuples of exponents separately. Under density and correlation assumptions on $\mathcal{A}$, we obtain a lower bound for the number of integers with many such representations, in terms of $d$ and $k$. In particular, when $\mathcal{A}$ is the set of primes or the set of positive integers representable as a sum of two squares, a positive proportion of the integers $n\le x$ have at least $c_1(\log x)^{d-1}$ or $c_1(\log x)^{d-1/2}$ representations, respectively, for some $c_1>0$ and all sufficiently large $x$. No multiplicative independence or coprimality assumptions on the bases are required.
