On several irrationality problems for Ahmes series

What's new 2024-11-28

Vjeko Kovac and I have just uploaded to the arXiv my paper “On several irrationality problems for Ahmes series“. This paper resolves (or at least makes partial progress on) some open questions of Erdös and others on the irrationality of Ahmes series, which are infinite series of the form ${\sum_{k=1}^\infty \frac{1}{a_k}}$ for some increasing sequence ${a_k}$ of natural numbers. Of course, since most real numbers are irrational, one expects such series to “generically” be irrational, and we make this intuition precise (in both a probabilistic sense and a Baire category sense) in our paper. However, it is often difficult to establish the irrationality of any specific series. For example, it is already a non-trivial result of Erdös that the series ${\sum_{k=1}^\infty \frac{1}{2^k-1}}$ is irrational, while the irrationality of ${\sum_p \frac{1}{2^p-1}}$ (equivalent to Erdös problem #69) remains open, although very recently Pratt established this conditionally on the Hardy–Littlewood prime tuples conjecture. Finally, the irrationality of ${\sum_n \frac{1}{n!-1}}$ (Erdös problem #68) is completely open.

On the other hand, it has long been known that if the sequence ${a_k}$ grows faster than ${C^{2^k}}$ for any ${C}$ , then the Ahmes series is necessarily irrational, basically because the fractional parts of ${a_1 \dots a_m \sum_{k=1}^\infty \frac{1}{a_k}}$ can be arbitrarily small positive quantities, which is inconsistent with ${\sum_{k=1}^\infty \frac{1}{a_k}}$ being rational. This growth rate is sharp, as can be seen by iterating the identity ${\frac{1}{n} = \frac{1}{n+1} + \frac{1}{n(n+1)}}$ to obtain a rational Ahmes series of growth rate ${(C+o(1))^{2^k}}$ for any fixed ${C>1}$ .

In our paper we show that if ${a_k}$ grows somewhat slower than the above sequences in the sense that ${a_{k+1} = o(a_k^2)}$ , for instance if ${a_k \asymp 2^{(2-\varepsilon)^k}}$ for a fixed ${0 < \varepsilon < 1}$ , then one can find a comparable sequence ${b_k \asymp a_k}$ for which ${\sum_{k=1}^\infty \frac{1}{b_k}}$ is rational. This partially addresses Erdös problem #263, which asked if the sequence ${a_k = 2^{2^k}}$ had this property, and whether any sequence of exponential or slower growth (but with ${\sum_{k=1}^\infty 1/a_k}$ convergent) had this property. Unfortunately we barely miss a full solution of both parts of the problem, since the condition ${a_{k+1} = o(a_k^2)}$ we need just fails to cover the case ${a_k = 2^{2^k}}$ , and also does not quite hold for all sequences going to infinity at an exponential or slower rate.

We also show the following variant; if ${a_k}$ has exponential growth in the sense that ${a_{k+1} = O(a_k)}$ with ${\sum_{k=1}^\infty \frac{1}{a_k}}$ convergent, then there exists nearby natural numbers ${b_k = a_k + O(1)}$ such that ${\sum_{k=1}^\infty \frac{1}{b_k}}$ is rational. This answers the first part of Erdös problem #263 which asked about the case ${a_k = 2^k}$ , althuogh the second part (which asks about ${a_k = k!}$ ) is slightly out of reach of our methods. Indeed, we show that the exponential growth hypothesis is best possible in the sense a random sequence ${a_k}$ that grows faster than exponentially will not have this property, this result does not address any specific superexponential sequence such as ${a_k = k!}$ , although it does apply to some sequence ${a_k}$ of the shape ${a_k = k! + O(\log\log k)}$ .

Our methods can also handle higher dimensional variants in which multiple series are simultaneously set to be rational. Perhaps the most striking result is this: we can find a increasing sequence ${a_k}$ of natural numbers with the property that ${\sum_{k=1}^\infty \frac{1}{a_k + t}}$ is rational for every rational ${t}$ (excluding the cases ${t = - a_k}$ to avoid division by zero)! This answers (in the negative) a question of Stolarsky Erdös problem #266, and also reproves Erdös problem #265 (and in the latter case one can even make ${a_k}$ grow double exponentially fast).

Our methods are elementary and avoid any number-theoretic considerations, relying primarily on the countable dense nature of the rationals and an iterative approximation technique. The first observation is that to represent a given number ${q}$ as an Ahmes series ${\sum_{k=1}^\infty \frac{1}{a_k}}$ for each ${a_k}$ lies in some interval ${I_k}$ (with the ${I_k}$ disjoint, and going to infinity fast enough to ensure convergence of the series), this is the same as asking for the infinite sumset

$\displaystyle \frac{1}{I_1} + \frac{1}{I_2} + \dots$

to contain ${q}$ , where ${\frac{1}{I_k} = \{ \frac{1}{a}: a \in I_k \}}$ . More generally, to represent a tuple of numbers ${(q_t)_{t \in T}}$ indexed by some set ${T}$ of numbers simultaneously as ${\sum_{k=1}^\infty \frac{1}{a_k+t}}$ with ${a_k \in I_k}$ , this is the same as asking for the infinite sumset

$\displaystyle E_1 + E_2 + \dots$

to contain ${(q_t)_{t \in T}}$ , where now

$\displaystyle E_k = \{ (\frac{1}{a+t})_{t \in T}: a \in I_k \}. \ \ \ \ \ (1)$

So the main problem is to get control on such infinite sumsets. Here we use a very simple observation:

Proposition 1 (Iterative approximatiom) Let ${V}$ be a Banach space, let ${E_1,E_2,\dots}$ be sets with each ${E_k}$ contained in the ball of radius ${\varepsilon_k>0}$ around the origin for some ${\varepsilon_k}$ with ${\sum_{k=1}^\infty \varepsilon_k}$ convergent, so that the infinite sumset ${E_1 + E_2 + \dots}$ is well-defined. Suppose that one has some convergent series ${\sum_{k=1}^\infty v_k}$ in ${V}$ , and sets ${B_1,B_2,\dots}$ converging in norm to zero, such that
$\displaystyle v_k + B_k \subset E_k + B_{k+1} \ \ \ \ \ (2)$
for all ${k \geq 1}$ . Then the infinite sumset ${E_1 + E_2 + \dots}$ contains ${\sum_{k=1}^\infty v_k + B_1}$ .

Informally, the condition (2) asserts that ${E_k}$ occupies all of ${v_k + B_k}$ “at the scale ${B_{k+1}}$ “.

Proof: Let ${w_1 \in B_1}$ . Our task is to express ${\sum_{k=1}^\infty v_k + w_1}$ as a series ${\sum_{k=1}^\infty e_k}$ with ${e_k \in E_k}$ . From (2) we may write

$\displaystyle \sum_{k=1}^\infty v_k + w_1 = \sum_{k=2}^\infty v_k + e_1 + w_2$

for some ${e_1 \in E_1}$ and ${w_2 \in B_2}$ . Iterating this, we may find ${e_k \in E_k}$ and ${w_k \in B_k}$ such that

$\displaystyle \sum_{k=1}^\infty v_k + w_1 = \sum_{k=m+1}^\infty v_k + e_1 + e_2 + \dots + e_m + w_{m+1}$

for all ${m}$ . Sending ${m \rightarrow \infty}$ , we obtain

$\displaystyle \sum_{k=1}^\infty v_k + w_1 = e_1 + e_2 + \dots$

as required. $\Box$

In one dimension, sets of the form ${\frac{1}{I_k}}$ are dense enough that the condition (2) can be satisfied in a large number of situations, leading to most of our one-dimensional results. In higher dimension, the sets ${E_k}$ lie on curves in a high-dimensional space, and so do not directly obey usable inclusions of the form (2); however, for suitable choices of intervals ${I_k}$ , one can take some finite sums ${E_{k+1} + \dots E_{k+d}}$ which will become dense enough to obtain usable inclusions of the form (2) once ${d}$ reaches the dimension of the ambient space, basically thanks to the inverse function theorem (and the non-vanishing curvatures of the curve in question). For the Stolarsky problem, which is an infinite-dimensional problem, it turns out that one can modify this approach by letting ${d}$ grow slowly to infinity with ${k}$ .