2 research outputs found
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"
A Single Set Improvement to the Theorem
The Theorem is a classical result which asserts that if
are finite, nonempty subsets with
\begin{equation}\label{hyp}|A+B|=|A|+|B|+r\leq
|A|+|B|+\min\{|A|,\,|B|\}-3-\delta,\end{equation} where if and
are translates of each other, and otherwise , then there are
arithmetic progressions and of common difference such that
, , and . It is one of the few cases in Freiman's Theorem for which exact
bounds on the sizes of the progressions are known. The hypothesis above is best
possible in the sense that there are examples of sumsets having
cardinality just one more, yet and cannot both be contained in short
length arithmetic progressions. In this paper, we show that the hypothesis
above can be significantly weakened and still yield the same conclusion for one
of the sets and . Specifically, if , is the unique
integer with
and \begin{equation}\label{hyp2} |A+B|=|A|+|B|+r<
(\frac{|A|}{s}+\frac{|B|}{2}-1)(s+1),\end{equation} then we show there is an
arithmetic progression with and
. The above hypothesis is best possible (without additional
assumptions on ) for obtaining such a conclusion