A proof that h_1, h_2 <= h_0 for any h-basis A_3

Abstract

A_k = {1, a_2, ... a_k} is an h-basis for X if every positive integer not exceeding X can be expressed as the sum of no more than h values a_i; X(h) is called the h-range of the basis. h_0 is the smallest value of h for which X(h) is greater than or equal to a_k, and h_1 is the smallest value for which X(h+1) = X(h) + a_k for all h greater than or equal to h_1. h_2 identifies a further "stabilisation" in the h-range - a definition is included in the body of the paper. It is known that h_1 and h_2 do not exceed h_0 for h-bases A_3, but published proofs are complicated (see Ch. VIII of [4], where references [3] and [5] are given). This paper introduces the concept of a "stride generator" A = {1, a_2, a_3} which, while sharing some of the properties of a basis A_3, is simpler to treat mathematically. We establish a relationship between stride generators and h-bases, and show that the result follows immediately if the stride generator underlying a basis has a particular property - here called "canonicality". The proof is lengthy (with a number of special cases to consider), but the underlying principles remain simple.