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Claim. The single Théorème (p. 765) of P. Erdős, Sur l'irrationalité d'une certaine série, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 17, 765--768, states that if $a_1<a_2<\cdots$ are integers with $a_{n+1}-a_n\to\infty$ then

$$\sum_{n\ge1}\frac{a_n}{2^{a_n}}$$

is irrational, and that the same holds with $2$ replaced by any integer $t>1$. Gaps tending to infinity force $a_n/n\to\infty$, so every such sequence is an instance of Problem 260, answered yes. The proof is by contradiction: if the sum were $u/(v2^r)$ with $v$ odd, then for each large $k$, let $a_n=2^k-s$ be the largest term below $2^k$ and write $a_{n+j}=2^k+t_j$. Multiplying through by $v2^{a_n}$ makes the tail in the note's equation (2) a positive integer, hence at least $1$. If $t_1+s>k$ (case (3)), the fractional-part estimate (4) bounds a tail sum below by $1/(2v^2)$, while (5) and (6) show that this sum tends to $0$, so case (3) fails for all large $k$. If $t_1+s\le k$, (7) and (8) write (2) as an integer plus three sums, two of which tend to $0$, so (9) forces the fractional part of the first sum towards $1$. Equations (10) to (13) show that this fractional part stays below $1-\delta$ for an absolute constant $\delta>0$, a contradiction. The note calls this case its only difficult part. Erdős writes that he had conjectured the theorem more than twenty years earlier, that before it only the case $a_n>cn\log n$ was known (his reference [2]), and that $\lim a_n/n=\infty$ very likely suffices, though he could not construct a sequence with $\limsup(a_{n+1}-a_n)=\infty$ and rational sum. The source card is erdos_1981_sur_l_irrationalite_d_une_certaine; the paper link is the author-archive scan of the note. The site's remarks credit the note with a second condition, $a_n\gg n\sqrt{\log n\log\log n}$; the note does not contain it, and the source of that condition is unidentified.

Covers. Every increasing integer sequence with $a_{n+1}-a_n\to\infty$, for the base $2$ of the question and for every integer base $t>1$. Not covered: sequences with $a_n/n\to\infty$ whose gaps do not tend to infinity, which is the gap the problem's question concerns and which Wang and Grau Ribas's pending claim addresses.

Acceptance. Refereed: the Comptes Rendus de l'Académie des Sciences, Série I, volume 292, number 17, the issue of 11 May 1981, pp. 765--768, as the note's header and its received line print. The site labels the problem OPEN, so its curator's remark crediting the note with this case is commentary on an open problem and not acceptance, and no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.