Abstract

Let \(0<t_1\le t_2\le t_3\le\dotsm\) be a given sequence of real numbers, and let \(\rho(t):=\sum_{j\colon t_j\le t}1\) be its counting function. We assume that \(\rho(t)=\rho_0t^\gamma(1+o(1))\) as \(t\to\infty\) for some \(\rho_0,\gamma>0\). For any \(T>0\) and \(k\in\overline{\mathbb{N}}=\mathbb{N}\cup\{\infty\}\), let \(N(k,T)\) be the number of solutions \(\{n_j\}_{j=1}^k\), \(n_j\in \mathbb{N}_0=\mathbb{N}\cup\{0\}\), of the inequality \(\sum_{j=1}^{k}n_jt_j\le T\). We are interested in the asymptotics of \(\log N(k,T)\) as \(T\to\infty\).

中文翻译：

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限制分区数的渐近性

摘要

令\（0 <t_1 \ le t_2 \ le t_3 \ le \ dotsm \）为给定的实数序列，并令\（\ rho（t）：= \ sum_ {j \冒号t_j \ le t} 1 \ ）作为其计数功能。我们假设\（\ RHO（T）= \ rho_0t ^ \伽马（1 + O（1））\）作为\（T \到\ infty \）对于一些\（\ rho_0，\伽马> 0 \） 。对于任何\（T> 0 \）和\（k \ in \ overline {\ mathbb {N}} = \ mathbb {N} \ cup \ {\ infty \} \），让\（N（k，T） \）是解的数量\（\ {n_j \} _ {j = 1} ^ k \），\（n_j \ in \ mathbb {N} _0 = \ mathbb {N} \ cup \ {0 \} \ ）中的不等式\（\ sum_ {j = 1} ^ {k} n_jt_j \ le T \）。我们对\（\ log N（k，T）\）的渐近性感兴趣为\（T \ to \ infty \）。