Duverney 2001 irrationality fast converging series rational numbers

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corollary_3_2: For positive integers u_n tending to infinity whose relative errors u_(n+1)/u_n^2 minus 1 form a convergent series, and signs a_n of plus or minus one, the sum of a_n/u_n is rational exactly when u_(n+1) equals u_n^2 minus (a_(n+1)/a_n)u_n plus a_(n+2)/a_(n+1) for all large n.

theorem_3_1: If a series of terms a_n/(b_n u_n) with u_(n+1) between two constant multiples of u_n^2, numerators of size O(u_n^alpha) with alpha below 1/7 and denominators b_n of subpolynomial size has a rational sum, then u_n eventually satisfies a quadratic recurrence whose leading coefficients p_n/q_n approximate u_(n+1)/u_n^2 and depend only on u_n.

theorem_3_2: When u_(n+1) equals beta u_n^2 plus O(u_n^gamma) with gamma below 2, the numerators and denominators grow like exp(o(2^n)), beta has irrationality exponent at most lambda and, for rational beta, the exceptional recurrence fails for large n, the series of a_n/(b_n u_n) has irrationality measure at most 4(2 lambda + omega)/omega.

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Duverney, Daniel, Irrationality of fast converging series of rational numbers. J. Math. Sci. Univ. Tokyo 8 (2001), 275--316.

Source: https://www.ms.u-tokyo.ac.jp/journal/abstract/jms080206.html. No copyright or license line is printed on pp. 275--276 or 315--316; the journal's article page (https://www.ms.u-tokyo.ac.jp/journal/abstract/jms080206.html, read 2026-10-02) shows only the site footer "©copyright 2013 Graduate School of Mathematical Sciences, The University of Tokyo All rights reserved.", which speaks for the website, and no per-article copyright or license statement; the term is unstated.

Digest

The paper studies series $\sum_{n\ge0}a_n/(b_nu_n)$ of nonzero rationals in which $u_{n+1}$ is of the order of $u_n^2$, so the terms decay doubly exponentially, and proves irrationality criteria by a weak form of Mahler's method in the form of Loxton and van der Poorten. Section 2 (pp. 277--285) surveys earlier results of Sylvester, Lucas, Golomb, Erdős and Straus, Badea, Hančl and the author; Section 3 (pp. 285--291) states the new results, which are proved in Sections 4--6 (pp. 291--314).

• Theorem 3.1 (pp. 285--286), the main result: under the growth conditions (1.3) with $\alpha<1/7$, a rational sum forces $u_n$ to satisfy, for all large $n$, a quadratic recurrence whose leading coefficients $p_n/q_n$ approximate $u_{n+1}/u_n^2$; the recurrence is also sufficient. Proved in Section 4 (pp. 291--298), the proof itself ending on p. 297.
• Corollary 3.2 (p. 287): for positive integers $u_n\to+\infty$, $a_n=\pm1$ and relative errors $u_{n+1}/u_n^2-1$ forming a convergent series (the print writes the sum as $<\infty$), $\sum a_n/u_n$ is rational exactly when a signed form of the Sylvester recurrence holds for all large $n$; the paper's partial answer to Erdős's question (2.15) on p. 280. Proved in Section 5.2, pp. 299--300.
• Theorem 3.2 (pp. 290--291): an irrationality measure $\tau=4(2\lambda+\omega)/\omega$, $\omega=\inf(2-\gamma,1)$, when $u_n\to+\infty$, $u_{n+1}=\beta u_n^2+O(u_n^{\gamma})$ with real $\beta>0$ and $0\le\gamma<2$, $\log|a_n|$ and $\log|b_n|$ are $o(2^n)$, $\beta$ satisfies the non-Liouville condition (3.18) with an exponent $\lambda\ge2$, and, if $\beta$ is rational, the recurrence of Corollary 3.4 fails for every $n\ge N$ (3.19). Proved in Section 6, pp. 311--314.

The other corollaries of Section 3 have no pages here. Corollary 3.1 (p. 287) shows, under the growth conditions (3.5), that $\sum_n r^n/u_n$ is irrational for every nonzero rational $r$ except perhaps one; Corollary 3.3 (p. 287) extends the Erdős--Straus theorem on $\sum1/(a^{2^n}+b_n)$ to the alternating case; Corollary 3.4 (p. 288) is a rationality criterion when $u_{n+1}=\beta u_n^2+O(u_n^{\gamma})$ with real $\beta>0$, $0\le\gamma<2$ and $\log|a_n|,\log|b_n|=o(2^n)$: the sum is rational if and only if $\beta$ is rational and the recurrence (3.9) holds; the print attaches no range of $n$ to (3.9), and the proof (p. 302) obtains it for $n\ge N_1(\alpha)$; Corollary 3.5 (p. 288) treats linear recurrences sampled at the indices $2^n$; Corollary 3.6 (pp. 289--290) refines Corollary 3.1 when $u_{n+1}=\beta u_n^2+O(u_n^{\gamma})$ with $\beta>0$ and $0<\gamma<2$. They are proved in Sections 5.1 and 5.3--5.6 (pp. 298--310).

Read depth. Claims checked for the three result pages above: their statements were read clause by clause in the printed article and their proofs for structure only. The other corollaries are summarized from their printed statements, not checked clause by clause.

Bears on

• Problem 243: Corollary 3.2 with every $a_n=1$ decides the problem for the sequences whose relative errors $u_{n+1}/u_n^2-1$ form a convergent series, and says nothing about sequences whose relative error tends to $0$ without forming one.

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