Nguyen (2022): Sparse Fibonacci reciprocal series

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theorem_1_2: Nguyen's transcendence theorem: if c > 2 and the positive integers n_1 < n_2 < ... satisfy n_{k+1}/n_k >= c for every k, then the sum of 1/f_k is transcendental for any choice of f_k among F_{n_k} and L_{n_k}.

theorem_1_3: Nguyen's main theorem: for a real quadratic unit alpha, rational terms u_n = a_n alpha^n - b_n beta^n with coefficients of subexponential height, and indices with n_{k+1}/n_k >= c > 2, the sum of c_{n_k}/u_{n_k} is transcendental.

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Khoa Dang Nguyen, Transcendental series of reciprocals of Fibonacci and Lucas numbers, Algebra & Number Theory 16 (2022), no. 7, 1627--1654, doi:10.2140/ant.2022.16.1627. The result labels below were checked in arXiv:2009.02446v1, rather than the publisher's text.

Question 1.1 (p. 1) repeats the Erdős--Graham question in Problem 267, which asks whether $\sum_k1/F_{n_k}$ is irrational whenever $n_{k+1}/n_k\ge c>1$. The paper's results:

• Theorem 1.2 (p. 1): if $c>2$ and $n_1<n_2<\cdots$ are positive integers with $n_{k+1}/n_k\ge c$ for every $k$, then

$$\sum_{k=1}^{\infty}\frac{1}{f_k}$$

is transcendental for any choice of $f_k\in\{F_{n_k},L_{n_k}\}$, where $L_n$ is the Lucas sequence: $L_1=1$ and $L_n=F_{n-1}+F_{n+1}$ for $n\ge2$. The print's definition reads $L_n=F_{n-1}+F_n$, a misprint, as Example 1.4's $L_n=\alpha^n+\beta^n$ shows.

• Theorem 1.3 (p. 3), the main theorem: the same conclusion for $\sum_kc_{n_k}/u_{n_k}$, where $\alpha\ne\pm1$ is a real quadratic unit with conjugate $\beta$, $|\beta|<1<|\alpha|$; $a_n,b_n,c_n$ are real with $c_n\in\mathbb Q$, $a_n,b_n\in\mathbb Q(\alpha)$ and $u_n=a_n\alpha^n-b_n\beta^n\in\mathbb Q$; their logarithmic Weil heights are $o(n)$; and $u_{n_k}\ne0$, $c_{n_k}\ne0$ for every $k$. Theorem 1.2 is its special case (Example 1.4, p. 3). The proof (Sections 3--5, pp. 5--26) uses the Subspace Theorem.

The paper points out (p. 2) that irrationality for $c>2$ already follows from the estimate $F_{n_1}\cdots F_{n_N}=o(F_{n_{N+1}})$; the new conclusion is transcendence. The abstract states that the bound $c>2$ is best possible, because of the identity $\sum_{k\ge0}1/F_{2^k}=(7-\sqrt5)/2$ (the Millin series, p. 1), whose index ratios equal $2$. Theorem 1.2 says nothing about sequences whose ratios are bounded below only by a constant $c\le2$.

Source: https://doi.org/10.2140/ant.2022.16.1627.

Edition read. The copy read for this card is the arXiv v1 PDF (arXiv:2009.02446v1). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2009.02446), every other right reserved.

Bears on. #267: Theorem 1.2 with $f_k=F_{n_k}$ gives transcendence, hence irrationality, of $\sum_k1/F_{n_k}$ when the ratios $n_{k+1}/n_k$ are at least a constant $c>2$; it does not reach constants $c\le2$.

Read status. Claims checked: Question 1.1, Theorems 1.2 and 1.3, Examples 1.4 and 1.5, and the comparison with the elementary irrationality bound were read in arXiv v1, pp. 1--3. The proof was read for its structure only and was not checked.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.