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Claim. Theorem 1 of the note "A Short Proof for Erdos Problem 1096" by Pedro Acosta De León (two pages, dated 16 April 2026, open access under a CC BY 4.0 license): with $\theta$ the smallest Pisot number, the real root of $x^3-x-1$, the gaps $x_{n+1}-x_n$ of the ordered sequence of finite sums of distinct powers of $q$ tend to $0$ for every $q\in(1,\sqrt\theta)$, so the question of Problem 1096 has the answer yes with $\epsilon=\sqrt\theta-1\approx0.151$. The proof is the one-paragraph deduction also made on the problem page: for such $q$ one has $q<\sqrt2$ and $q^2<\theta$, so $q^2$ is not a Pisot number and Feng's Theorem 1.4 gives $L_1(q)=0$, the upper limit of the gaps, hence the gaps tend to $0$. The note's Remark 1 says the argument does not reach the larger interval $(1,\theta)$. It adds no mathematics beyond Feng's theorem and Siegel's theorem on the smallest Pisot number, and its range is narrower than the range Erdős and Komornik's Theorem IV had already settled in 1998.

Posting. The note was posted on a data repository on 16 April 2026 and linked the same day from the first comment of the problem's thread, whose poster calls it an unverified proof; it was never filed on the site's proof-claims tab, and the curator's reply in the thread credits Feng and then Erdős and Komornik without mentioning the note. No review, referee report or other acceptance evidence exists, so the claim is pending. Read depth: the whole note; its deduction is the one checked on the problem page.

Depends on. Feng's Theorem 1.4 on Feng's page; the deduction also uses Siegel's theorem on the smallest Pisot number, cited on the problem page as [Si44].