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Source. Theorem 5.1 and Remark 5.1, preprint p. 11, with the opening of Section 5 (p. 11); Theorem 5.2 (p. 11, proof p. 12), Examples 5.1--5.2 (p. 12) and Example 5.3 (p. 13) for the related constructions. Read on the rendered pages. The paper is cited by its record on the source card.

Statement

For a sequence $(b_n)_{n\ge1}$ and a positive integer $k$ put $T_N=\sum_{n\ge N}b_nk^{N-n}$ ($N\ge1$). Let $k>1$ be an integer and $(b_n)_{n\ge1}$ a sequence of positive integers such that $T=\sum_{n\ge1}b_nk^{-n}$ converges and $b_n\le(1-1/k)T_{n+1}$ for all $n$. Let $S\in(T/(k+1),T]$. Then there are $a_n\in\{k,k+1,\ldots,k^2\}$ with

$$S=\sum_{n=1}^{\infty}\frac{b_n}{a_1\cdots a_n}.$$

Remark 5.1 (p. 11): every nondecreasing sequence of positive integers $b_n$ with $T$ convergent satisfies the condition, so the theorem applies to all such sequences. The opening of Section 5 records that Hančl and Tijdeman had the case $k=2$, $a_n\in\{2,3,4\}$, for nondecreasing $b_n$ and $S\in(T/2,T)$. The proof chooses $a_n$ greedily from the position of the current remainder $S_n$ relative to $T_n$ and sets $S_{n+1}=a_nS_n-b_n$.

Related constructions

• Theorem 5.2 (p. 11): for an integer $k>1$ and positive integers $b_n$ with $T=\sum b_nk^{-n}$ convergent and $T_{N+1}\ge(k+1)b_N$ for $N>1$, every $S\in(k^2T/(k+1)^2,T]$ is $\sum b_n/(a_1\cdots a_n)$ with $a_n\in\{k,k+1\}$; Example 5.1 (p. 12) gives $b_n=[(k-\frac13)^n]$ as a sequence meeting these conditions.
• Example 5.2 (p. 12): with $b_n=(n-2)!$ for $n\ge2$, every $S\in(\frac{146}{75},\frac83]$ is $\sum_{n\ge2}b_n/(a_2\cdots a_n)$ with $a_n\in\{n,n+1\}$; the paper calls this in some sense a counterpart to Theorems 3.1, 4.1 and 4.2.
• Example 5.3 (p. 13): for integers $d>c>1$, $0<\epsilon<(cd-c)/(d^2-c)$ and $b_n=(d-1)^n$, every $S\in\bigl(c(d-1)(d-1+\epsilon)/(d^2\epsilon),(d-1)/\epsilon\bigr]$ is $\sum b_n/(a_1\cdots a_n)$ with $a_n\in\{c,d\}$.

The paper closes Section 5 (p. 13) with two questions it calls open: for an integer $k\ge2$, positive integers $b_n$ with $\sum b_nk^{-n}$ convergent and distinct integers $a,b\ge k$, whether there is a fixed interval each of whose points is $\sum b_n/(a_1\cdots a_n)$ with every $a_n\in\{a,b\}$, and whether there are infinitely many such representations.

Role

The paper presents these constructions (p. 2) as showing that the results of Sections 3 and 4 do not hold without growth restrictions. For instance, with $k=2$ and $b_n=p_n$, the $n$th prime, Remark 5.1 applies, so every number in $(T/3,T]$, where $T=\sum p_n/2^n$, is $\sum p_n/(a_1\cdots a_n)$ for some $a_n\in\{2,3,4\}$; this specialization is the corpus's, and it says nothing about the rationality of $T$ itself.

Bears on. No catalog problem directly.