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Claim. J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set, Acta Sci. Math. (Szeged) 21 (1960), 111–124, received 3 September 1959 (the page name's date). With $A(n)$ the number of elements of $A$ up to $n$, its Theorem I (p. 111) states: if $(A(2n)-A(n))/\log\log n\to\infty$ and $\sum_{a\in A}\|a\theta\|^2=\infty$ for every real $\theta$ with $0<\theta<1$, then every sufficiently large integer is a sum of distinct elements of $A$. Both hypotheses imply those of Problem 254: the first gives $A(2n)-A(n)\to\infty$, hence $|A\cap[1,2x]|-|A\cap[1,x]|\to\infty$ along real $x$, and since $\|x\|\le1/2$ we have $\|a\theta\|^2\le\|a\theta\|/2$, so the second gives $\sum_{a\in A}\|a\theta\|=\infty$. The proof (Section 2) applies the circle method to a thinned subset of $A$; the paper's digest is the library card.

Covers. The sets $A$ that satisfy both of Cassels's hypotheses; for them the conclusion of Problem 254 holds. Not covered: a set for which $(A(2n)-A(n))/\log\log n$ does not tend to infinity, or for which $\sum_{a\in A}\|a\theta\|^2$ converges at some $\theta\in(0,1)$; the problem asks for the conclusion under its weaker hypotheses.

Acceptance. Refereed: Acta Scientiarum Mathematicarum (Szeged) 21 (1960). The site's commentary credits Cassels with the statement under these hypotheses, but the site labels the problem OPEN, so no reviewed evidence is listed.

Depends on. No page of this wiki.