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Source. Theorem 3.3 and its proof, preprint p. 12. Read on the rendered page. The edition read is identified on the source card.

Statement

Let $(a_n)_{n\ge1}$ be a sequence of Gaussian integers such that for infinitely many $N$ the terms $a_N,a_{N+1},\ldots,a_{4N}$ form a geometric sequence with $|a_{N+1}/a_N|\le2$. Assume that

$$a_n=o\bigl(n^{n/7}\bigr)\qquad(15)$$

for $n$ sufficiently large. Then $S=\sum_{n=1}^{\infty}a_n/n!\notin\mathbb{Q}[i]$.

Proof pointer

By the proof of Proposition 3.2 (see Theorem 3.2) it suffices that $D_N\ne0$. With $a=1$, $b=0$ the main term of $D_N$ is $qc^Na_{2N}\frac{N!(2N-1)!}{(3N)!}$ times $1$ plus a series whose modulus the bound $|c/d|\le2$ keeps below $(1+o(1))(e^{2/3}-1)<0.95$; hence $|D_N|>\frac1{20}\cdot7^{-N}$ for large $N$ (p. 12).

Dependencies

Proposition 3.2 and Lemma 2.3 of the same paper.

Bears on

No catalog problem directly.