For a real set $A$ consider the semigroup ${\mathcal S}(A)$, additively generated by $A$; that is, the set of all real numbers representable as a (finite) sum of elements of $A$. If $A\subseteq(0,1)$ is open and non-empty, then ${\mathcal S}(A)$ is easily seen to contain all sufficiently large real numbers, and we let $G(A):=\sup\{u\in{\mathbb R}: u\notin{\mathcal S}(A)\}$. 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 $\alpha:= {\rm mes\,} A$, we have

\[ G(A) \le \begin{cases} (1-\alpha)\lfloor 1/\alpha\rfloor &\text{if $0<\alpha\le 0.1$}, \\ (1-\alpha+\alpha\{1/\alpha\})\lfloor 1/\alpha\rfloor &\text{if $0.1 \le \alpha \le 0.5$}, \\ 2(1-\alpha) &\text{if $0.5 \le \alpha \le 1$}. \\ \end{cases} \]

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 Erdös and Graham), also known as the ‘postage stamp problem’ or the ‘coin exchange problem’.