  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".
