---

Claim. Theorem 1.4 of Akiyama and Komornik (arXiv v1, p. 4) is printed as "Let $1<q<2$ be a non-Pisot number. (i) If $1<q\le\sqrt[3]{2}\approx1.26$, then $L_1(q)=0$", where $L_1(q)=\limsup(x_{n+1}-x_n)$ for the increasing sequence $0=x_0<x_1<\cdots$ (the paper's indexing) of the sums $\sum s_iq^i$ with digits $s_i\in\{0,1\}$, the sequence of Problem 1096; parts (ii) and (iii) give $\ell_1(q)=L_2(q)=0$ for $1<q\le\sqrt2$ and $\ell_2(q)=L_3(q)=0$ for $1<q<2$, with $\ell_m$ and $L_m$ the lower and upper limits of the gaps for digits $0,\ldots,m$. Every $q$ in $(1,2^{1/3}]$ lies below the smallest Pisot number $q_0\approx1.3247$, so the non-Pisot hypothesis holds throughout part (i), as the paper's proof notes, and $x_{k+1}-x_k\to0$ for every $1<q\le2^{1/3}\approx1.2599$: the problem's question has the answer yes with $\epsilon=2^{1/3}-1\approx0.26$. The paper says that part (i) improves Theorem IV of Erdős and Komornik, which had $L_1(q)=0$ for $1<q\le2^{1/4}$ except possibly $\sqrt{P_2}$, the square root of the second Pisot number; that point lies in the new range, so this theorem closes it. The proof (Section 5) treats $q=2^{1/3}$ by adapting the proof of a proposition of an earlier paper it cites; for $1<q<2^{1/3}$ with $q^3$ not Pisot it combines the paper's main Theorem 1.1 (the difference set $Y^m(q)$ of the sums with digits $0,\ldots,m$ has a finite accumulation point exactly when $q<m+1$ and $q$ is not Pisot) with its Lemma 5.3 (an accumulation point of $Y^m(q^3)$ gives $L_m(q)=0$); when $q^3$ is Pisot it shows $\ell_1(q^2)=0$ through a theorem of Sidorov and Solomyak on the conjugates of $q^2$ and passes to $L_1(q)=0$ by its Lemma 2.5. Feng's 2016 paper (p. 3) reports the result in the same form, $L_1(q)=0$ for $1<q\le2^{1/3}$, cited to Erdős and Komornik and to this paper together.

Source. S. Akiyama and V. Komornik, Discrete spectra and Pisot numbers, J. Number Theory 133 (2013), no. 2, 375--390, DOI 10.1016/j.jnt.2012.07.015 (Crossref record: issue dated February 2013, record created 12 October 2012); arXiv:1103.4508v1 of 23 March 2011, the text whose page is cited. The journal text is not compared, and the paper is not held in the library.

Acceptance. Refereed: the paper appeared in the Journal of Number Theory. The site's commentary credits the problem's resolution to Erdős and Komornik and to Feng and does not mention this paper, so no reviewed evidence is listed. Read depth: claims checked for Theorem 1.4 and for the structure of its proof in Section 5 of the arXiv text; Theorem 1.1 and the lemmas are taken at their statements and not checked. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki. The same answer, for narrower ranges, is on Erdős and Komornik's page and Feng's page; none of the three rests on another.