The continuous postage stamp problem

TL;DR

This paper investigates the continuous analogue of the Frobenius coin problem, establishing bounds on the largest non-representable number based on the measure of the set, with precise results for certain measure ranges.

Contribution

It provides new bounds on the Frobenius number for continuous sets, extending classical discrete results to a continuous setting with optimal and near-optimal estimates.

Findings

For large measure sets, the Frobenius number is small.

Exact bounds are given for measures up to 0.5.

The bounds are optimal or near-optimal for specific set configurations.

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".