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On the irrationality of certain super-polynomially decaying series

Tonći Crmarić, Vjekoslav Kovač

TL;DR

The paper settles Erdős and Graham's question by proving that the sums of the form $\sum_{n=1}^{\infty} \frac{1}{\prod_{i=1}^{f(n)}(n+i)}$ with $f(n)\to\infty$ can realize every positive real number, not just irrational values, so the set of possible sums is $(0,\infty)$. It achieves this via Kakeya-type analysis of subsums for convergent positive series and a two-step constructive framework that densely fills target values, while also highlighting that when $f(n)$ is assumed to be increasing the problem becomes substantially more delicate. The authors extend Kakeya's ideas to general finite-sets $X_n$, proving a dichotomy for the resulting sumsets (finite union of intervals versus closed sets with empty interior) and providing a detailed, constructive argument. They further show that the increasing-$f(n)$ variant yields a Lebesgue-measure-zero, fractal-like set, indicating that monotonicity imposes strong topological constraints and complicates the irrationality question in this regime.

Abstract

We give a negative answer to a question by Paul Erdős and Ronald Graham on whether the series \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)\cdots(n+f(n))} \] has an irrational sum whenever $(f(n))_{n=1}^{\infty}$ is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when $(f(n))_{n=1}^{\infty}$ is additionally assumed to be increasing.

On the irrationality of certain super-polynomially decaying series

TL;DR

The paper settles Erdős and Graham's question by proving that the sums of the form with can realize every positive real number, not just irrational values, so the set of possible sums is . It achieves this via Kakeya-type analysis of subsums for convergent positive series and a two-step constructive framework that densely fills target values, while also highlighting that when is assumed to be increasing the problem becomes substantially more delicate. The authors extend Kakeya's ideas to general finite-sets , proving a dichotomy for the resulting sumsets (finite union of intervals versus closed sets with empty interior) and providing a detailed, constructive argument. They further show that the increasing- variant yields a Lebesgue-measure-zero, fractal-like set, indicating that monotonicity imposes strong topological constraints and complicates the irrationality question in this regime.

Abstract

We give a negative answer to a question by Paul Erdős and Ronald Graham on whether the series has an irrational sum whenever is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when is additionally assumed to be increasing.

Paper Structure

This paper contains 4 sections, 4 theorems, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary observations
  3. Proof of Theorem \ref{['thm:main']}
  4. Proof of Theorem \ref{['thm:cantor']}

Key Result

Theorem 1

The set is equal to the whole interval $(0,\infty)$.

Figures (2)

  • Figure 1: Partial series of \ref{['eq:generalseries']} with $1\leqslant f(1)\leqslant 5$, $2\leqslant f(2),f(3)\leqslant 5$, $3\leqslant f(4),f(5),f(6),f(7)\leqslant 5$; the first $3000$ sums sorted in the ascending order.
  • Figure 2: Partial series of \ref{['eq:generalseries']} with $1\leqslant f(1)\leqslant f(2)\leqslant f(3)\leqslant f(4)\leqslant f(5)\leqslant f(6)\leqslant f(7)\leqslant 8$; the first $3000$ sums sorted in the ascending order.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Lemma 3: Kakeya
  • Lemma 4
  • proof
  • Remark 5
  • Remark 6