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Claim. Theorem 1.4 of Feng (p. 3 of arXiv v3; the journal's Theorem 1.4) is printed as "If $1<q<\sqrt{m+1}$ and $q^2$ is not a Pisot number, then $L_m(q)=0$. In particular, if $q\in(1,\sqrt2)$ and $q^2$ is not a Pisot number, then $L_1(q)=0$", where $L_m(q)$ is the upper limit of the consecutive gaps of the ordered sums of powers of $q$ with digits $0,\ldots,m$; for $m=1$ the sequence is the $0=x_1<x_2<\cdots$ of Problem 1096 and $L_1(q)=0$ says $x_{k+1}-x_k\to0$. The paper derives the theorem in one line from its Corollary 1.3 ($\ell_m(q)=0$, the lower limit of the gaps, exactly when $q<m+1$ and $q$ is not Pisot, a corollary of the main density theorem) and the implication $\ell_m(q^2)=0\Rightarrow L_m(q)=0$, Lemma 2.5 of Akiyama and Komornik, first proved for $m=1$ as Theorem 5 of the 1998 Acta Arithmetica paper of Erdős, Joó and Komornik. The result first appears in arXiv v2 as Proposition 3.1 of Section 3.2, for general $m$, with the $m=1$ case stated there for $q\in(2^{1/3},\sqrt2)$ and the range $(1,2^{1/3}]$ credited to Akiyama and Komornik; v3 and the journal print it as Theorem 1.4 in the form quoted above. arXiv v1 states neither: its Theorem 1.4 concerns finite accumulation points of $Y_m(q)$ and it does not define $L_m(q)$. The deduction to the problem: the smallest Pisot number is $q_0\approx1.3247$, the real root of $x^3=x+1$ (Siegel's theorem of 1944, which the site's commentary also states), so no Pisot number lies in $(1,q_0)$; for $1<q<\sqrt{q_0}$ one has $q^2\in(1,q_0)$, hence $q^2$ is not Pisot, and $q<\sqrt{q_0}<\sqrt2$, so Theorem 1.4 with $m=1$ gives $x_{k+1}-x_k\to0$. Since $\sqrt{q_0}\approx1.1510$, every $\epsilon\le\sqrt{q_0}-1\approx0.151$ answers the question: yes. Feng's paper does not state this consequence, as the site's curator observes in the thread; the deduction is the one paragraph above, made on the problem page and in the curator's thread post. The theorem is compiled on the result page theorem_1_4; the digest is on the card feng_2016_topology_polynomials_bounded_integer_coefficients.

Scope of the theorem. Theorem 1.4 decides every $q\in(1,\sqrt2)$ except the square roots of Pisot numbers ($\sqrt{q_0}\approx1.1510$, $\sqrt{q_1}\approx1.175$, and so on), where it is silent; the earlier Theorem IV on [[problems/number_theory/E1096/claims/1998_04_01_erdos_komornik|Erdős and Komornik's page]] covers $\sqrt{q_0}$ by a separate case and leaves $\sqrt{q_1}$ out below $2^{1/4}$, a point closed by Theorem 1.4 (i) on Akiyama and Komornik's page, which gives $L_1(q)=0$ for every $1<q\le2^{1/3}$. The three pages give the same answer by routes through $q^2$ or $q^3$, and none rests on another.

Acceptance. Refereed: De-Jun Feng, On the topology of polynomials with bounded integer coefficients, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 1, 181--193, whose Crossref record dates the article 16 December 2015; the page is named by the first arXiv posting that carries the claim, 1109.1407v2 of 10 November 2011 (as Proposition 3.1; v1 of 7 September 2011 does not contain it), and the text cited is v3 of 1 February 2015, not compared with the journal text. Reviewed: the site's curator, Thomas F. Bloom, wrote in the problem's thread on 16 April 2026 that the problem was solved by Feng, through Theorem 1.4 and the gap between $1$ and the smallest Pisot number, before finding the earlier resolution of Erdős and Komornik, and the problem's commentary states both of Feng's results; the curator neither wrote nor submitted the result. Read depth: claims checked for Theorem 1.4, Corollary 1.3, Theorem 1.2 and the derivation sentence; Feng's proof (Theorem 1.6 by way of Theorem 1.11, pp. 5--6, and Section 2, pp. 6--11) and the cited implication are not checked, and nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki; the deduction uses Siegel's theorem on the smallest Pisot number, cited on the problem page as [Si44], a classical fact that is not compiled here.