Improved bounds on the postage stamp problem for large numbers of stamps

TL;DR

This paper improves bounds on the minimal size of additive bases covering all numbers up to n with h elements, especially for large h, using probabilistic and combinatorial methods.

Contribution

It provides the first nontrivial asymptotic upper bound and refines the lower bound on the size of h-fold bases for large h.

Findings

Improved asymptotic bounds on F_h(n) for large h.

First nontrivial upper bound on F_h(n).

Enhanced understanding of additive bases in number theory.

Abstract

Let $F_h(n)$ denote the minimum cardinality of an additive {\em $h$-fold basis} of $\{1,2,\cdots,n\}$: a set $S$ such that any integer in $\{1,2,\cdots, n\}$ can be written as a sum of at most $h$ elements from $S$. While the trivial bounds $h!n \; \lesssim \; F_h(n)^h \; \lesssim \; h^h n$ are well-known, comparatively little has been established for $h>2$. In this paper, we make significant improvements to both of the best-known bounds on $F_h(n)$ for sufficiently large $h$. For the lower bound, we use a probabilistic approach along with the Berry-Esseen Theorem to improve upon the best-known asymptotic result due to Yu. We also establish the first nontrivial asymptotic upper bound on $F_h(n)$ by leveraging a construction for additive bases of finite cyclic groups due to Jia and Shen. In particular, we show that given any $\epsilon>0$, for sufficiently large $h$, we have \[ \left(\frac{1}{2}-\epsilon\right)h!\sqrt{2\pi e} n\; \leq \; F_h(n)^h \; \leq \; \left(\left(\frac{\sqrt{3}}{2}+\epsilon\right)h\right)^h n. \]