A note on m_h(A_k)

TL;DR

This paper investigates properties of h-bases in the Postage Stamp Problem, introduces the concept of admissible sets, and presents a counterexample to a conjecture about the pattern of missing values.

Contribution

It provides a counterexample to Selmer's conjecture on the non-increasing pattern of missing values differences in h-bases.

Findings

Counterexample found for Selmer's conjecture

Analysis of the pattern of missing values in h-bases

Insights into the structure of extremal h-bases

Abstract

A_k = {1, a_2, ..., a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i; we write n = n_h(A_k). An extremal h-basis A_k is one for which n is as large as possible, and then we write n = n_h(k). The "local" Postage Stamp Problem is concerned with properties of particular sets A_k, and it is clear that sets where n_h(A_k) does not exceed a_k are of little interest. We define h_0(k) to be the smallest value of h for which n_h(A_k) exceeds a_k; such sets are called "admissible". We say that a value n can be "generated" by A_k if it can be expressed as the sum of no more than h values a_i, or - equivalently - if it can be expressed as the sum of exactly h values a_i from the set A'_k = {0, a_1, a_2, ... a_k}. No values greater than ha_k can be generated, and we now consider the number of values less than ha_k that have no generation, denoted m_h(A_k) - essentially a count of the number of "gaps" (see Challis [1], and Selmer [5] page 3.1). It is easy to show that for some value h_2(k) exceeding h_0(k) the difference m_h(A_k) - m_(h+1)(A_k) remains constant - that is, the "pattern" of missing values between ha_k and (h+1)a_k does not change as h increases. Here we are interested in the pattern of missing values for values that lie between h_0 and h_2. On page 7.8 of Selmer [5] he conjectures that the sequence of differences m_h(A_k) - m_(h+1)(A_k) is non-increasing as h runs from h_0 to h_2. When I came across this conjecture I could not convince myself that it was likely to be true, having found a possible error in Selmer's justification. I wrote to him in November 1995, and early in 1996 he replied, agreeing that this might be the case and hoping that I might be able to find a counter example. This paper records my successful search for a counter example, eventually found late in 1999.