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The continuous postage stamp problem
For a real set consider the semigroup , additively generated by
; that is, the set of all real numbers representable as a (finite) sum of
elements of . If is open and non-empty, then is
easily seen to contain all sufficiently large real numbers, and we let . Thus, is the smallest number
with the property that any is representable as indicated above.
We show that if the measure of is large, then 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 and
, respectively; the second is close to
the best possible and can be improved by 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"
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