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Source. Theorem 1, p. 1 of the arXiv PDF; proof pp. 2--3. Proposition 1, p. 1; proof pp. 3--4. The proof of Theorem 1 uses the quoted Lemma 1 (Weyl–van der Corput, p. 2). Read on the rendered pages.

Statement

For a real number $\lambda\ge0$ put

$$S_\lambda=\sum_{n\ge0}\frac{[n^\lambda]}{n!},$$

$[\cdot]$ the integer part. Then the set

$$\{1,e\}\cup\{S_\lambda:\lambda\in(0,\infty)\setminus\mathbb{Z}\}$$

is $\mathbb{Q}$-linearly independent.

The paper presents this as an explicit set of uncountably many $\mathbb{Q}$-linearly independent real numbers (p. 1). Integer values of $\lambda$ are excluded; for $\lambda=1$, for instance, the series is $\sum_{n\ge1}1/(n-1)!=e$.

Proposition 1 (p. 1). The map $\lambda\mapsto S_\lambda$ is injective, monotone and continuous from the right. Its image has the cardinality of the continuum, has Hausdorff dimension $0$, and is totally disconnected.

Proof structure

Theorem 1 (pp. 2--3). It suffices to show that, for nonzero integers $a_1,\ldots,a_k$ and exponents $0\le\lambda_1<\cdots<\lambda_k$ none of which is an integer $\ge2$, at least one of them nonzero, the number $\sum_{n\ge0}\big(a_1[n^{\lambda_1}]+\cdots+a_k[n^{\lambda_k}]\big)/n!$ is irrational. If it equals $p/q$, then for $n\ge q$ the scaled tail is an integer; truncating it at $\nu=[\lambda_k]+1$ and dropping the integer parts costs $O(1/n)$, giving formula (1) (p. 3): the distance to the nearest integer of a finite sum $f(n)$ is $\ll1/n$. If $\lambda_k<1$ this is contradicted directly. If $\lambda_k>1$, the non-integrality of $\lambda_k$ gives $K\in\mathbb{N}$ such that $f^{(K+1)}(t)$ and $f^{(K+2)}(t)$ do not change sign for $t>t_0$, $f^{(K)}(t)/t\to0$ and $1/(tf^{(K+1)}(t))\to0$ as $t\to\infty$, so $f(n)$ is equidistributed modulo $1$ (Lemma 1 and Hlawka's book), which contradicts (1).

Proposition 1 (pp. 3--4). Injectivity: for $\lambda_2>\lambda_1$ and large $n$ the integer parts differ by at least $1$. Right continuity: a tail bound on $S_t-S_\lambda$ for $t$ slightly above $\lambda$. Total disconnectedness follows from Theorem 1, since an interval of positive length in the image would contain infinitely many rationals. Hausdorff dimension $0$: the partial sums up to $N$ take at most $N^{t+2}$ values on $[t,t+1]$, and the tail is below $N^{-A}$ for any fixed $A$.

Where scrutiny would begin

Recorded for a future review; none has been made. The proof treats the cases $\lambda_k<1$ and $\lambda_k>1$; the reduction admits $\lambda_k=1$ (an integer below $2$), which is not treated separately in the text.

Bears on. No catalog problem directly.