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Statement

As printed on p. 389, among the paper's closing open problems, with $0=:y_1<y_2<\ldots$ the increasing sequence of finite sums of distinct nonnegative powers of $q$ (p. 386):

Problem 4: "Characterize the set of those $1<q<2$ for which $y_{n+1}-y_n\to0$. Is it true that every $q$ which is sufficiently close to $1$ has this property?"

The second sentence is the question of Problem 1096 (whether there is $\epsilon>0$ such that $x_{k+1}-x_k\to0$ for every $1<q<1+\epsilon$), with the problem's $x_k$ equal to the paper's $y_k$. The site's other source for the problem is the 1991 problem session of Great Western Number Theory (not held), where the site says Erdős and Joó posed it and speculated that the threshold may be the smallest Pisot number $q_0\approx1.3247$; that speculation is not printed here. Theorem 4 d) of the same paper shows the first sentence's set is not all of $(1,A)$, $A=(1+\sqrt5)/2$, and Remark 2 that it contains every $2^{1/m}$, $m\ge2$.

Source. P. Erdős, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum_{i=1}^\infty q^{-n_i}$ and related problems, Bull. Soc. Math. France 118 (1990), 377--390; Problem 4 on printed p. 389 (PDF p. 14 of the Numdam file), read on the rendered page image. The edition is identified in the source digest.

Read depth. Claims checked: the problem and its neighbors (Problems 1--6, pp. 389--390) were read clause by clause on the page images. A question, not a theorem; nothing to verify.

Proof pointer

None (an open problem as posed). Its second sentence was answered affirmatively by Erdős and Komornik's 1998 paper (the site's ErKo98, filed as theorem_iv) and, with an explicit threshold, by Feng's Theorem 1.4 (theorem_1_4), as recorded on the problem page. The first sentence asks for the set of $q$ where the upper limit of the gaps is $0$; Feng's Corollary 1.3 characterizes the $q$ where the lower limit is $0$, and the problem page records the characterization for the upper limit as open in general by Feng's account (p. 3).

Dependencies

None.

Bears on

• Problem 1096: the problem's question in the authors' own words, printed in 1990; the site's source key EJK90 for the problem and its source for the gap bound and the Pisot obstruction (Theorem 4).