The Postage Stamp Problem and Essential Subsets in Integer Bases

TL;DR

This paper extends the analysis of essential elements in additive bases by studying the asymptotic behavior of the maximum number of essential subsets of size k, linking it to the postage stamp problem.

Contribution

It introduces the asymptotic analysis of E(h,k), generalizing previous work on essential elements, and connects the problem to the postage stamp problem in finite cyclic groups.

Findings

E(h,k) = Θ_k([h^k / log h]^{1/(k+1)}) as h→∞ for fixed k

E(h,k) ∼ (h-1) log k / log log k as k→∞ for fixed h

The precise asymptotics depend on solving the postage stamp problem.

Abstract

Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h,k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that E(h,k) = \Theta_{k} ([h^{k}/\log h]^{1/(k+1)}). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known "postage stamp problem" in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show that E(h,k) \sim (h-1) {\log k \over \log \log k}.