THE CONTINUOUS POSTAGE STAMP PROBLEM

For a real set A consider the semigroup S(A), additively generated by A; that is, the set of all real numbers representable as a (finite) sum of elements of A. If A ⊆ (0, 1) is open and non-empty, then S(A) is easily seen to contain all sufficiently large real numbers, and we let G(A) := sup{u ∈ R : u /∈ S(A)}. Thus G(A) is the smallest number with the property that any u > G(A) is representable as indicated above. We show that if the measure of A is large, then G(A) is small; more precisely, writing for brevity α := mesA, we have G(A)   (1 − α) 1/α if 0 < α 0.1, (1 − α + α{1/α}) 1/α if 0.1 α 0.5, 2(1 − α) if 0.5 α 1. Indeed, the first and the last of these three estimates are the best possible, attained for A = (1−α, 1) and A = (1−α, 1)\{2(1−α)}, respectively; the second is close to the best possible and can be improved by α{1/α} 1/α {1/α} at most. The problem studied is a continuous analogue of the linear Diophantine problem of Frobenius (in its extremal settings due to Erd˝os and Graham), also known as the ‘postage stamp problem’ or the ‘coin exchange problem’.