The postage stamp problem and arithmetic in base r

Abstract

Let h, k be fixed positive integers, and let A be any set of positive integers. Let hA ≔ {a 1 + a 2 + ... + a r : a i ∈ A, r ⩽ h} denote the set of all integers representable as a sum of no more than h elements of A, and let n(h, A) denote the largest integer n such that {1, 2,...,n} ⊆ hA. Let n(h, k) := $$ \mathop {\max }\limits_A $$ : n(h, A), where the maximum is taken over all sets A with k elements. We determine n(h, A) when the elements of A are in geometric progression. In particular, this results in the evaluation of n(h, 2) and yields surprisingly sharp lower bounds for n(h, k), particularly for k = 3.