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Source. P. Erdős, Sur l'irrationalité d'une certaine série, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 17, 765--768 (French translation by J.-L. Nicolas). The note's single Théorème, unnumbered, is stated on p. 765; its proof runs from p. 765 to p. 767. Bibliographic details are on the source card.

Statement

Let $a_1<a_2<\cdots$ be a sequence of integers with $a_{n+1}-a_n\to+\infty$. Then

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

is irrational.

The theorem as printed reads: "Soit $a_1<a_2<\ldots$ une suite d'entiers satisfaisant $a_{n+1}-a_n\to+\infty$. Alors $\sum_{n\ge1}a_n/(2^{a_n})$ est irrationnel." (p. 765).

In the paragraph after the statement Erdős adds, without proof, that the theorem remains true when $2$ is replaced by an integer $t>1$ (p. 765); the proof given is written for base $2$ only. He also records that he had conjectured the result more than twenty years earlier and that only the case $a_n>cn\log n$ had been known, citing problem 180 of Wiskundige Opgaven met de Oplossingen 20 (1955--1959) (p. 765).

Proof sketch (pp. 765--767)

Suppose the sum equals $u/(v2^r)$ with $r\ge0$, $v$ odd and $(u,v2^r)=1$. For large $k$, let $a_n=2^k-s$ be the largest term below $2^k$ and write the following terms as $2^k+t_1,2^k+t_2,\ldots$. Multiplying by $v2^{a_n}$ makes the tail sum (2) times $v$ a positive integer, hence at least $1$.

• If $t_1+s>k$ for infinitely many $k$ (case (3)), the oddness of $v$ gives the lower bound (4) of $1/(2v^2)$ on a tail which, by (5) and (6) and the growth of the gaps $t_{j+1}-t_j$, tends to $0$; so (3) fails for all large $k$.
• Otherwise $t_1+s\le k$; with $i$ defined by (7), the integer equation (8) splits into three sums whose total (9) is a positive integer. The second and third sums tend to $0$, so the first must have fractional part tending to $1$, and the fractional-part estimate (13), with an absolute constant $\delta>0$, rules this out (p. 767).

This sketch is written from a reading of the proof's structure; the estimates were not re-derived here.

Read depth. Claims checked: the statement was read clause by clause on p. 765 of the printed note; the proof was read for structure only.

Dependencies

None beyond elementary estimates; the note cites no lemma.

Bears on

• Problem 260: the problem asks the same question under the hypothesis $a_n/n\to\infty$. Gaps tending to infinity imply $a_n/n\to\infty$, so the theorem answers the question for the sequences whose gaps tend to infinity; it says nothing about other sequences with $a_n/n\to\infty$. The paper itself poses the weaker hypothesis as its conjecture; see the conjecture on p. 765.