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Source. T. Crmarić and V. Kovač, On the irrationality of certain super-polynomially decaying series, Colloquium Mathematicum (2025), doi:10.4064/cm9628-5-2025; arXiv:2504.18712v1 (25 April 2025). Theorem 2 on p. 3 of the arXiv v1 PDF; its proof is Section 4 (pp. 9--11). Bibliographic details and reading limits are in the source card.

Statement

With $\mathbb N$ the positive integers, the theorem states that the set

$$\Bigl\{\ \sum_{n=1}^{\infty}\frac{1}{\prod_{i=1}^{f(n)}(n+i)}\ :\ (f(n))_{n=1}^{\infty}\in\mathbb N^{\mathbb N}\text{ is increasing},\ \lim_{n\to\infty}f(n)=\infty\Bigr\}$$

(the paper's (1.4)) has zero Lebesgue measure and, consequently, empty interior.

"Increasing" is used in the non-strict sense: the proof on p. 10 treats every $f$ with $f(1)\le f(2)\le\cdots$, and the tuples it counts on p. 9 satisfy $t_1\le t_2\le\cdots\le t_{m-1}$. The theorem does not say whether the set contains a rational number. The authors conclude (p. 3) that an easy negative answer should no longer be expected in this case, since finding a rational number in a negligible set is hard.

Proof sketch (Section 4, pp. 9--11)

Fix $M\in\mathbb N$, choose $K$ with $\sum_{n=1}^{K}1/(n+1)>M$ (the paper's (4.1)) and let $N\ge K$. Each admissible $f$ determines the first index $m$ with $f(m)>N$ and the nondecreasing tuple $f(1),\dots,f(m-1)\le N$; its sum lies in an interval whose left end is fixed by that tuple and whose length, $\frac1N\prod_{i=1}^{N}(m+i)^{-1}$, depends only on $N$ and $m$. The union (4.2) of these intervals covers the set inside $[0,M]$. Counting the $\binom{m+N-2}{N-1}$ tuples gives total length below $1/N!$ for $m\le N$ ((4.3)); for $m>N$ only tuples not beginning with $K$ ones can meet $[0,M]$, and their total length is below $2^N/(N-2)!$ ((4.4)). Both bounds tend to $0$ as $N\to\infty$, so each intersection with $[0,M]$ is null, and so is the union over $M$.

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. 3 of the arXiv v1 PDF, and the reading of "increasing" against p. 10.

Dependencies

None beyond elementary counting and the divergence of the harmonic series; the proof follows the covering argument of [irrationality/crmaric_2025_irrationality_certain_super_polynomially_decaying_series/lemma_4|Lemma 4] and Remark 6 without applying the lemma itself.

Bears on

• Problem 270: context for the variant with nondecreasing $f$, which the problem's statement does not impose; the theorem shows the values form a null set and neither proves nor disproves irrationality in that case.