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 $\alpha := \mes A$ we have $$ G(A) \le (1-\alpha) \lfloor 1/\alpha \rfloor \quad &\text{if $0 < \alpha \le 0.1$}, (1-\alpha+\alpha\{1/\alpha\})\lfloor 1/\alpha\rfloor \quad &\text{if $0.1 \le \alpha \le 0.5$}, 2(1-\alpha) \quad &\text{if $0.5 \le \alpha \le 1$}. $$ Indeed, the first and the last of these three estimates are the best possible, attained for $A=(1-\alpha,1)$ and $A=(1-\alpha,1)\setminus\{2(1-\alpha)\}$, respectively; the second is close to the best possible and can be improved by $\alpha \{1/\alpha\} \lfloor 1/\alpha \rfloor \le \{1/\alpha\}$ 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".