Notes on Symmetric Bases

TL;DR

This paper investigates symmetric bases in the context of the Postage Stamp Problem, proposing a conjecture about coverage ranges and demonstrating that it holds beyond a certain threshold h_1.

Contribution

It introduces a conjecture on symmetric bases' coverage properties and shows that the conjecture is true for all h greater than or equal to a specific value h_1.

Findings

The conjecture is not strictly true in general.

There exists a threshold h_1 beyond which the conjecture holds.

Symmetric bases have specific properties influencing coverage ranges.

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. An extremal h-basis A_k is one for which n is as large as possible. Computing extremal bases is known as the Postage Stamp Problem. A basis A_k is symmetric if A_k = {1, a_2, ... a_k} where a_i + a_(k-i) = a_k for 1<=i<=k-1. Examination of a number of symmetric bases suggests the following conjecture: if the range 0 ... a_k is covered using at most h stamps, then the range 0 ... ha_k is also covered using at most h stamps. This paper shows that this is not strictly true, but demonstrates that there is a value h_1 such that the conjecture is true for all h>=h_1.