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Source. Theorem 2, printed p. 215, physical PDF p. 4; proof pp. 218--219; the surrounding remarks on p. 213. Read on the page images.

Statement

Fix integers $t>1$ and $l\ge1$, and let $1<n_1<n_2<\cdots$ be integers with $\limsup_{k\to\infty}n_k/k^l=\infty$. Then

$$\alpha=\sum_{k=1}^{\infty}\frac{1}{t^{n_k}}$$

is a root of no nonzero polynomial with integer coefficients of degree at most $l$. (For $l=1$ this says that $\alpha$ is irrational.)

Context on p. 213

The paper first recalls a result with Straus: if $\limsup\log n_k/\log k=\infty$, then $\sum_k1/t^{n_k}$ is transcendental (the footnote line reads "Elemente der Math. 9, 18 Problem 154, (1954)"). Theorem 2 is described as obtained "by a modification of our method used there". Then: "I do not know to what extent this theorem can be improved, I do not know if a series $\sum_{k=1}^{\infty}\frac{1}{t^{n_k}}$ satisfying $\limsup n_k/k=\infty$ can be an algebraic number. On the other hand I cannot even prove that if $n_k>ck^2$ then $(\sum_{k=1}^{\infty}\frac{1}{t^{n_k}})^2$ is always irrational."

Structure of the proof (pp. 218--219)

Assume (20): $d_0\alpha^{l_1}+d_1\alpha^{l_1-1}+\cdots+d_{l_1}=0$ with integers $d_i$, $d_0>0$ and $1\le l_1\le l$.

• One may assume (21) $n_{k+1}<c_5n_k$ for all $k$: otherwise $\limsup n_{k+1}/n_k=\infty$, and $\alpha-\sum_{i\le k}t^{-n_i}<2t^{-n_{k+1}}=2\,(t^{-n_k})^{n_{k+1}/n_k}$ makes $\alpha$ a Liouville number, hence transcendental, contradicting (20).
• Expanding by the multinomial theorem, $d_0\alpha^{l_1}=\sum_ka_k/t^k$ and $d_1\alpha^{l_1-1}+\cdots+d_{l_1}=\sum_k\varepsilon_kb_k/t^k$ with nonnegative integers $a_k$, $b_k$ and signs $\varepsilon_k$; here $a_k>0$ exactly when $k$ is a sum of $l_1$ terms $n_i$, and $b_k>0$ only when $k$ is a sum of fewer than $l_1$ terms.
• The hypotheses of Lemma 4 are checked: (5) holds with $s=l_1+1$; choosing $k_i$ with $n_{k_i}/k_i^{\,l}\to\infty$ and $m_i=n_{k_i}$, the counts satisfy $f(n_{k_i})\le k_i^{l_1}=o(n_{k_i})$, $g(n_{k_i})\le k_i^{l_1-1}=o(n_{k_i}^{1-1/l_1})$ and $\sum_{j\le n_{k_i}}(a_j+b_j)<c_5k_i^{l_1}=o(n_{k_i})$, which is (6); for (C), if $b_k>0$ then $k$ is a sum of $r<l_1$ terms, so $k+(l_1-r)n_i$ carries a positive $a$ for every $i$, and (21) supplies the constant $c_2$.
• Lemma 4 then makes $\sum_k(a_k+\varepsilon_kb_k)/t^k$, the left side of (20), irrational, contradicting (20).

These steps were read for structure and are recorded as a sketch; no complete rewritten proof and no independent review exist here.

Relation to Problem 247

Problem 247 asks whether $\sum_n1/2^{a_n}$ is transcendental whenever $1\le a_1<a_2<\cdots$ and $\limsup a_n/n=\infty$. The p. 213 question quoted above is that problem for a general integer base $t$, and Theorem 2 is its partial result: under $\limsup n_k/k^l=\infty$ the sum satisfies no integer equation of degree at most $l$, which for the hypothesis of Problem 247 ($l=1$) gives irrationality only. Transcendence under $\limsup n_k/k=\infty$ is left open by the paper.

Bears on. #247, as a partial result and a 1957 statement of the question.