The continuous postage stamp problem

Abstract

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 $A \subset (0,1)$ is open and non-empty, then $S(A)$ is easily seen to contain all sufficiently large real numbers, and we let $G(A) := \sup \{u \in R \colon u \notin S(A) \}$. 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 $α:= \mes A$ we have $$ G(A) \le (1-α) \lfloor 1/α\rfloor \quad &\text{if $0 < α\le 0.1$}, (1-α+α\{1/α\})\lfloor 1/α\rfloor \quad &\text{if $0.1 \le α\le 0.5$}, 2(1-α) \quad &\text{if $0.5 \le α\le 1$}. $$ Indeed, the first and the last of these three estimates are the best possible, attained for $A=(1-α,1)$ and $A=(1-α,1)\setminus\{2(1-α)\}$, respectively; the second is close to the best possible and can be improved by $α\{1/α\} \lfloor 1/α\rfloor \le \{1/α\}$ at most. The problem studied is a continuous analogue of the linear Diophantine problem of Frobenius (in its extremal settings due to Erdos and Graham), also known as the "postage stamp problem" or the "coin exchange problem".