Problem 1096

Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$ ), ordered by size as $0=x_1<x_2<\cdots$ .

Is it true that, provided $\epsilon>0$ is sufficiently small, $x_{k+1}-x_k \to 0$ ?

Source: erdosproblems.com/1096

Number theory

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SolvedProved

An accepted solution exists. The statement is true.

The site's label is PROVED (LEAN), whose catalog suffix is explained under Formalization. The first resolution is Theorem IV of Erdős and Komornik's 1998 paper (Acta Math. Hungar. 79 (1998), 57--83, refereed), an accepted full claim on its page: "If $1<q\le2^{1/4}$ and if $q$ is different from the square root of the second Pisot number, then $y_{k+1}-y_k\to0$ for every $m\ge1$ ", where $(y_k)$ is the ordered sequence of the sums with digits $0,\ldots,m$ and $m=1$ gives the problem's sequence; its introduction (p. 57) names the question as [EJK90]'s Problem 4 and says "One of the purposes of this paper is to give an affirmative answer to this question". With $2^{1/4}\approx1.1892$ and $\sqrt{q_1}\approx1.1749$ ( $q_1\approx1.38$ , the second Pisot number, the paper's $p_2$ ), every $q$ in $(1,\sqrt{q_1})$ is covered, so the answer is yes, with $\epsilon=\sqrt{q_1}-1\approx0.175$ , and so is every $q$ in $(\sqrt{q_1},2^{1/4}]$ . Theorem 1.4 (i) of Akiyama and Komornik (J. Number Theory 133 (2013), 375--390, refereed; quoted from the arXiv text), the second accepted full claim, on their page, gives $L_1(q)=0$ for every $1<q\le2^{1/3}\approx1.2599$ , a range that contains the excluded point $\sqrt{q_1}$ and so closes it. A third first-hand source, the third accepted full claim, on Feng's page, is Theorem 1.4 of Feng's paper (J. Eur. Math. Soc. 18 (2016), 181--193, refereed; quoted from arXiv v3): for $1<q<\sqrt2$ with $q^2$ not a Pisot number, $\lim(x_{n+1}-x_n)=0$ . Since $q_0\approx1.3247$ , the real root of $x^3=x+1$ , is the smallest Pisot number (Siegel's classical theorem, which the site's commentary also states), every $q$ in $(1,\sqrt{q_0})$ has $q^2\in(1,q_0)$ not Pisot and $q<\sqrt2$ , so the gaps tend to $0$ for every $1<q<\sqrt{q_0}\approx1.1510$ , with $\epsilon=\sqrt{q_0}-1\approx0.151$ (an authored deduction, one line, from Feng's theorem and Siegel's theorem). The site reports the 1998 range as $1<q<\sqrt{q_1}$ and Feng's introduction as $1<q\le2^{1/4}$ "with the possible exception of the square root of the second Pisot number"; the printed theorem is Feng's version, and the site's range is its part below the excluded point. The frontmatter standing is derived from the three accepted pages; a fourth, pending claim, the thread's two-page note on Acosta De León's page, repeats the deduction from Feng's theorem and adds nothing to the standing.

EJK90: Erdős, Pál and Joó, István and Komornik, Vilmos, Characterization of the unique expansions $1=\sum^\infty_{i=1}q^{-n_i}$ and related problems. Bull. Soc. Math. France 118 (1990), no. 3, 377--390, doi:10.24033/bsmf.2151 (Crossref record accessed). Theorem 4, p. 386; Problem 4, p. 389. Library home: Erdős, Joó and Komornik 1990.
GWNT91: The site's key for "the 1991 problem session of Great Western Number Theory". Located: problem 91:18 (Paul Erdős & I. Joó), p. 16, of Western Number Theory Problems, 1991-12-19 & 22, edited by Richard K. Guy; not held. Library home: Guy 1991.
Bu96: Bugeaud, Y., On a property of Pisot numbers and related questions. Acta Math. Hungar. 73 (1996), no. 1--2, 33--39, doi:10.1007/BF00058941 (Crossref bibliographic query). Not held (publisher paywall). Its theorem is quoted from the site and from [EJS96], p. 95: for $1<q<2$ , $q$ is Pisot if and only if $l_k(q)>0$ for all $k\ge1$ .
EJS96: Erdős, P. and Joó, I. and Schnitzer, F. J., On Pisot numbers. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 39 (1996), 95--99 (received 6 October 1995; no Crossref record). The Theorem, p. 95, in the journal archive's volume file. Library home: Erdős, Joó and Schnitzer 1996.
ErKo98: Erdős, P. and Komornik, V., Developments in non-integer bases. Acta Math. Hungar. 79 (1998), no. 1--2, 57--83, doi:10.1023/A:1006557705401 (Crossref record accessed; received 30 September 1996, per p. 83). Abstract and introduction, p. 57; Theorem IV with its remarks, pp. 59--60; its proof, pp. 77--78. Feng's reference [9]. Library home: Erdős and Komornik 1998.
Fe16: Feng, De-Jun, On the topology of polynomials with bounded integer coefficients. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 1, 181--193, doi:10.4171/JEMS/587 (Crossref record accessed); arXiv:1109.1407v3 (1 February 2015, "to appear in J. Eur. Math. Soc"; the journal text not compared). Theorem 1.2, p. 2; Corollary 1.3 and Theorem 1.4, p. 3. Library home: Feng 2016.
EJK98: Erdős, P., Joó, I. and Komornik, V., On the sequence of numbers of the form $\varepsilon_0+\varepsilon_1q+\ldots+\varepsilon_nq^n$ , $\varepsilon_i\in\{0,1\}$ . Acta Arith. 83 (1998), no. 3, 201--210, doi:10.4064/aa-83-3-201-210 (Crossref record accessed). Not a site key. Theorem 4, p. 206; Theorem 5, p. 207. Library home: Erdős 1998.
Si44: Siegel, C. L., Algebraic integers whose conjugates lie in the unit circle. Duke Math. J. 11 (1944), 597--602. The classical source of the fact that $q_0$ is the smallest Pisot number; not held, cited as the standard reference for a fact the site's commentary also states.

Problem 1096

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