Erdos 1998 sequence numbers form sums powers q

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theorem_4: The 1998 Erdős-Joó-Komornik bound L(q) <= (q^2 - 1)e on the upper limit of the consecutive gaps of the ordered finite sums of distinct powers of q in (1, 2), stated as the weaker result available because the authors did not know whether L(q) = 0 for all q sufficiently close to 1; the dated limitation behind Problem 1096.

theorem_5: The 1998 Erdős-Joó-Komornik implication that for 1 < q < sqrt 2 a vanishing lower gap limit for q^2 forces the gaps of the ordered sums of distinct powers of q to tend to 0, hence for every transcendental q below sqrt 2; the m = 1 case of the implication Feng's Theorem 1.4 uses for Problem 1096.

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Paul Erdős, István Joó and Vilmos Komornik, 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 (the Crossref record). "Received on 14.5.1996 and in revised form on 14.7.1997" (p. 210). Cited by the 1996 Erdős--Joó--Schnitzer paper as an IRMA Strasbourg preprint and by Feng (2016) as his reference [8]. The site's Problem 1096 page does not cite it; its keys ErKo98 and EJK90 are the authors' other papers.

The retained folder-name PDF is the journal's typeset file (pdfTeX, 2007), 10 pages, printed pp. 201--210 (printed p. $n$ is PDF p. $n-200$), with a text layer that drops the plus signs; the statements below were read on the rendered page images of pp. 201--202, 206 and 207. Provenance: retained from the repository's survey download set of 5 September 2026 (the download URL was not recorded; the journal's archive at https://www.impan.pl/ hosts the article under its DOI); 176,460 bytes. The file's text layer carries no copyright or license line; the publisher's record (https://www.impan.pl/get/doi/10.4064/aa-83-3-201-210, read 2026-10-02) offers the PDF under the link "Pobierz zgodnie z CC-BY" ("Free download under CC-BY license" on the English site), a Creative Commons Attribution license whose version the record does not name; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Read status: claims checked for the introduction (the definitions of $l(q)$ and $L(q)$, the results (a)--(f)), Theorem 1 as a statement, the sentence opening Section 3, Theorem 4, Theorem 5, Lemmas 6--8 and Proposition 9 as statements, read clause by clause on the page images of pp. 201--202 and 206--209; the proofs of Theorems 4 and 5 (pp. 206--209) were read for structure and not checked; Corollary 2, Proposition 3 and Lemma 6's proof were not checked.

Contents

• Section 1, Introduction (pp. 201--202). Fix $1<q<2$; for $k=\varepsilon_0+2\varepsilon_1+\cdots+2^n\varepsilon_n$ in binary set $x_k=\varepsilon_0+\varepsilon_1q+\cdots+\varepsilon_nq^n$, and let $y_0<y_1<\cdots$ be the increasing rearrangement of $(x_k)$ without repetitions, so $y_0=0$, $y_1=1$, $y_2=q$ and $y_k\to\infty$; $l(q)=\inf(y_{k+1}-y_k)$ and $L(q)=\limsup(y_{k+1}-y_k)$, with the remark that $l(q)=\liminf(y_{k+1}-y_k)$ (a small gap translated by a large power $q^n$ recurs arbitrarily far out). Recalled results, "the first three of them were proved in [3], while the last one was obtained in [2]" (p. 201; [3] the authors' 1990 Bulletin paper, [2] Erdős, Joó and Joó 1992): (a) $0\le l(q)\le L(q)\le1$ for all $1<q<2$; (b) $L(q)=1$ for all $A\le q<2$, $A=(1+\sqrt5)/2$; (c) $L(q)>0$ for all Pisot numbers; (d) $l(q)=1/q>0$ for the Pisot numbers with $q^{r+1}=1+q+\cdots+q^r$. New results announced: (e) $l(q)>0$ for all Pisot numbers (also obtained independently by Bugeaud [1], with a partial converse); (f) $L(q)=0$, i.e. $y_{k+1}-y_k\to0$, for all transcendental $1<q<\sqrt2$.
• Section 2, Pisot numbers (pp. 202--206): Theorem 1 ($l(q)>0$ for all Pisot numbers, with the estimates (3)--(5) in terms of $\sum_{k\ge N}\|q^k\|$), Corollary 2 (lower bounds for $L(q)$ and $l(q)$ through the conjugates of $q$, p. 204) and Proposition 3 ($L(q)\ge q-1$ for the fourth Pisot number, p. 205), not checked.
• Section 3, Numbers $q$ close to $1$ (pp. 206--209): "We do not know whether $L(q)=0$ for all $q$ sufficiently close to $1$. We have the following weaker result:" (p. 206) Theorem 4 (p. 206): $L(q)\to0$ as $q\to1$; more precisely $L(q)\le(q^2-1)e$ for all $1<q<2$. "Our next result shows that $y_{k+1}-y_k\to0$ for almost all numbers $q$ sufficiently close to 1." (p. 207) Theorem 5 (p. 207): if $1<q<\sqrt2$ and $l(q^2)=0$ then $L(q)=0$, i.e. $y_{k+1}-y_k\to0$; in particular for transcendental $1<q<\sqrt2$. Lemmas 6--8 and the proofs (pp. 207--209); Proposition 9 (p. 209), $L(\sqrt2)=0$.
• Correction (pp. 209--210) to the proof of Theorem 4 c) of the 1990 paper (two sentences at the bottom of its p. 388 and the top of p. 389 replaced; "The rest of the proof is the same"), with the generalization to every $x\in(0,1/(q-1))$.
• Section 4, Open problems (p. 210): 1. "Is it true that $l(q)>0$ if and only if $q$ is a Pisot number?" 2. The exact values of $l(q)$ and $L(q)$ for Pisot numbers.

Compiled scope

The introduction and Section 3's statements were read on the page images; Theorems 4 and 5 are compiled as statements with proof pointers. No step of a proof was checked and nothing here is independently reviewed.

Bears on. #1096: the same ordered sequence as the problem's (the paper's $y_k$), with the problem's question stated in the authors' words as unknown in 1998 ("We do not know whether $L(q)=0$ for all $q$ sufficiently close to 1", p. 206) and two partial results, Theorem 4 ($L(q)\le(q^2-1)e$, so the upper limit of the gaps tends to $0$ with $q$) and Theorem 5 (gaps tending to $0$ for every $q<\sqrt2$ whose square has $l(q^2)=0$, hence for every transcendental $q<\sqrt2$); Theorem 5 is also the $m=1$ case of the implication Feng's Theorem 1.4 uses, so it sits in the proof chain behind the problem's held answer. The recall (a) attributes the bound $L(q)\le1$, and (c) the Pisot obstruction, to the authors' 1990 paper.

Results.

• Theorem 4 (p. 206): $L(q)\le(q^2-1)e$ for all $1<q<2$, hence $L(q)\to0$ as $q\to1$.
• Theorem 5 (p. 207): $1<q<\sqrt2$ and $l(q^2)=0$ imply $L(q)=0$; in particular for transcendental $1<q<\sqrt2$.
• Theorem 1 (p. 202): $l(q)>0$ for all Pisot numbers (statement read; not compiled as a page).