Erdos komornik 1998 developments non integer bases

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theorem_i: Erdős and Komornik's characterization of the Pisot numbers by the set of finite sums of powers of q with integer digits of modulus at most m: no finite accumulation point for every m when q is Pisot, and one for every integer m >= q - 1/q when q is not; its Part (b) supplies the first hypothesis of Lemma 3.2 in the proof of Theorem IV.

theorem_ii: Erdős and Komornik's theorem on the smallest gaps of the ordered finite sums of powers of q with digits 0, ..., m: the lim inf of the gaps is positive for every m when q is a Pisot number, and zero for every m >= [q - 1/q]' + [q - 1]' (upper integer parts) when q is not.

theorem_iii: Erdős and Komornik's theorem that the consecutive gaps of the ordered finite sums of powers of q with digits 0, ..., m tend to zero for every non-Pisot q once m >= [q - 1/q]' + 2[q - 1]' (upper integer parts), a bound that is never as small as 1 or 2.

theorem_iv: Erdős and Komornik's theorem that the consecutive gaps of the ordered finite sums of powers of q with digits 0, ..., m tend to zero for every m when 1 < q <= 2^(1/4) and q is not the square root of the second Pisot number; the m = 1 case is the first resolution of Problem 1096, on a range whose two second-hand accounts the printed statement reconciles.

theorem_v: Erdős and Komornik's theorem that when the gaps of the ordered finite sums of powers of q with digits 0, ..., m tend to zero, every alpha in (0, m/(q - 1)) has a development in base q with digits 0, ..., m that contains every finite string of those digits as a block.

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P. Erdős and V. Komornik, Developments in non-integer bases, Acta Math. Hungar. 79 (1998), no. 1--2, 57--83, DOI 10.1023/A:1006557705401 (the Crossref record; the printed pages carry the journal header "Acta Math. Hungar. 79 (1--2) (1998), 57--83" and the footer "Acta Mathematica Hungarica 79, 1998" and no DOI). The byline prints the first author's name in a box, the journal's mark for a deceased author, "member of the Academy (Budapest)", and the second author at Strasbourg; the closing page gives the Mathematical Institute of the Hungarian Academy of Sciences and the Institut de Recherche Mathématique Avancée of the Université Louis Pasteur and CNRS as the two addresses. "Received September 30, 1996" (p. 83); the remarks on pp. 59--60 date the results to the authors' last discussions in July 1996 in Budapest and say the second author kept the paper in its state at those discussions after the first author's death. Cited as [ErKo98] on the problem page. The edition read for this card is the publisher's version of record; no preprint or repository version is known here. Its nine references (p. 83) include [4], the authors' 1990 Bulletin paper with Joó, filed as erdos_1990_characterization_unique_expansions_related_problems; [5], the Acta Arith. paper with Joó "to appear", filed as erdos_1998_sequence_numbers_form_sums_powers_q; [6], the 1996 Erdős--Joó--Schnitzer paper filed as erdos_1996_pisot_numbers; [7], Bugeaud, On a property of Pisot numbers and related questions, Acta Math. Hungar. 73 (1996), 33--39; [1], Bertin et al., Pisot and Salem numbers (Birkhäuser, 1992), cited for the list of Pisot numbers; [2], Bogmér, Horváth and Sövegjártó, On some problems of I. Joó, Acta Math. Hungar. 58 (1991), 153--155; [3], Erdős, Horváth and Joó, On the uniqueness of the expansions $1=\sum q^{-n_i}$, Acta Math. Hungar. 58 (1991), 333--342; [8], Frougny, Representations of numbers and finite automata, Math. Systems Theory 25 (1992), 37--60; and [9], Khintchine, Continued Fractions (1964).

The copy read for this card is the publisher's production PDF of the printed article: 27 pages, printed pp. 57--83 = PDF pp. 1--27 (printed p. $n$ is PDF p. $n-56$), A4 page images produced with Acrobat PDFWriter 3.0 (the file's metadata gives a creation date of 10 April 1998 and a modification date of 27 October 2004), with no text layer at all, so every passage below was read on the page images. Provenance: obtained from the publisher on 2026-09-22 as a DRM-free production PDF through the library's acquisition, the DOI https://doi.org/10.1023/A:1006557705401 resolving to the article's page at the publisher; 1,187,663 bytes. The file prints "0236-5294/98/$5.00 © 1998 Akadémiai Kiadó, Budapest" in the footer of its first page (read on the page image), every other right reserved.

Read status: claims checked for the abstract and § 0 (pp. 57--60), that is the definitions of the sequence $(y_k)$, the set $Y=Y^{q,m}$ and the sequences $y_k^{q,m}$, the attribution of the question to [4], Problem 4, Theorems I--V with their remarks, and the definition of a universal development, each read clause by clause on the page images of PDF pp. 1--4 on 2026-09-22; the proof of Theorem IV (pp. 77--78, PDF pp. 21--22) was read in full on the page images and its reductions to Part (b) of Theorem I and to Lemmas 3.1 and 3.2 were followed; the statements of Lemmas 1.1--1.8, 2.1--2.2, 3.1--3.2 and 4.1 and of Propositions 2.3, 3.3 and 3.4 were read on the page images, and the proofs of Theorems I, II, III and V, of Lemma 3.2 and of the propositions were read on the page images for structure only; none of them was checked. The reference list and the received date (p. 83, PDF p. 27) were read on the page image. On 2026-10-08 the statements of Theorems I, II, III and V and their remarks were read again clause by clause on the page images for their result pages, with the statements of Lemmas 1.1--1.8, 2.1--2.2, 3.1--3.2 and 4.1 and of Propositions 2.3, 3.3 and 3.4, and the one-paragraph proofs of Part (b) of Theorem II (p. 74) and of Theorem III (p. 77) were followed as reductions to their lemmas. Nothing here is independently reviewed.

Contents

• Abstract and § 0, Introduction (pp. 57--60, page images). Fix $q>1$ and let $0=y_0<y_1<y_2<\cdots$ be the increasing sequence of the real numbers with at least one representation $y=\varepsilon_0+\varepsilon_1q+\cdots+ \varepsilon_nq^n$, $n\ge0$, $\varepsilon_i\in\{0,1\}$. The sequence tends to infinity, more slowly as $q$ approaches 1, and the introduction (p. 57) names the first question, attributed to [4], Problem 4: do the gaps $y_{k+1}-y_k$ tend to 0 for every $q$ close enough to 1? The paper points out one difficulty: for some algebraic $q$ a number $y$ has several representations, while $(y_k)$ lists each value once. It then claims the answer, "One of the purposes of this paper is to give an affirmative answer to this question" (p. 57), and announces the special case of Theorem IV that delivers it, $y_{k+1}-y_k\to0$ for every $q$ between 1 and $2^{1/4}$ "except possibly the square root of the second Pisot number $\sqrt{p_2}\approx1.175$". A parenthetical note of the second author adds that the same property "probably holds also if $q=\sqrt{p_2}$" and that the first author's sudden death prevented the study. The second question (p. 58), from Bogmér, Horváth and Sövegjártó [2]: whether for each $q$ close to 1 some development $1=\sum_{i\ge1}\varepsilon_iq^{-i}$ contains arbitrarily long runs of the digit 0; Theorem IV with Part c) of Theorem 4 of [4] gives such developments for every $q$ between 1 and $2^{1/4}$ except possibly $\sqrt{p_2}$, and Theorem V gives developments containing every finite variation of the digits. For $q>1$ and an integer $m\ge1$, $Y=Y^{q,m}$ is the set of the sums $s_0+s_1q+\cdots+s_nq^n$ with $s_i\in\{0,\pm1,\ldots,\pm m\}$, and Pisot numbers are the algebraic integers $q>1$ all of whose conjugates lie in $|z|<1$. Theorem I (p. 58, quoted): "(a) If $q$ is a Pisot number, then $Y$ has no finite accumulation points for any $m$. (b) If $q$ is not a Pisot number, then $Y$ has finite accumulation points for every integer $m\ge q-q^{-1}$." Remark: the bound on $m$ is probably not sharp, and Proposition 2.3 shows that for $m\le(q-1)/2$ there are none, whatever $q$. Then $y_k^{q,m}$ is the increasing sequence of the sums with digits $\varepsilon_i\in {0,1,\ldots,m}$ (pp. 58--59), and $Y$ is the set of the differences $y_k-y_l$. Theorem II (p. 59, quoted): "(a) If $q$ is a Pisot number, then $\liminf(y_{k+1}-y_k)>0$ for every $m$. (b) If $q$ is not a Pisot number, then $\liminf(y_{k+1}-y_k)=0$ for every $m\ge[q-q^{-1}]'+[q-1]'$ where we denote by $[\alpha]'$ the upper integer part of $\alpha$ (i.e. the smallest integer $\ge\alpha$)." Remarks: Part (a) for $1<q<2$ was proved earlier by Bugeaud [7] by a different approach; Bugeaud also proved that a non-Pisot $1<q<2$ has some $m$ with $\liminf=0$ without an estimate, and Part (b) shows $m=3$ suffices there; Lemma 2.1 (b) gives $\liminf(y_{k+1}-y_k)=1$ whenever $m\le q-1$. The paper calls the $\limsup$ the harder quantity. Theorem III (p. 59, quoted): "If $q$ is not a Pisot number, then $y_{k+1}-y_k\to0$ for every $m\ge[q-q^{-1}]'+2[q-1]'$." Remark: Proposition 3.3 gives $\limsup(y_{k+1}-y_k)\ge1$ whenever $m\le q-q^{-1}$. The paper notes that Theorem III never applies for $m=1$ or $m=2$, so it leaves the first question open, which the next theorem settles. Theorem IV (p. 59, quoted): "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 $m\ge1$." Remarks (pp. 59--60): (a) the same property probably holds at $\sqrt{p_2}$, a question the authors meant to study during a visit in October 1996; (b) on the convergence rate, Proposition 3.4 shows an exponential rate for $q=\sqrt2$ and $m=1$. A development $\alpha=\sum_{i\ge1}\varepsilon_iq^{-i}$ with digits in $\{0,1,\ldots,m\}$ is universal if every finite variation $\delta_1\ldots\delta_k$ of the integers $0,1,\ldots,m$ occurs as a block of $(\varepsilon_i)$. Theorem V (p. 60, quoted): "Let $q$ and $m$ be such that $y_{k+1}-y_k\to0$. Then every $0<\alpha<m/(q-1)$ has a universal development." Combined with Theorem IV: for $1<q\le2^{1/4}$, $q\ne\sqrt{p_2}$, every $m$ and every $0<\alpha<m/(q-1)$ has a universal development, "the last joint result of the authors, obtained in July 1996". Remark (b): for fixed $m$, the bases $q$ in which $\alpha=1$ has a universal development form a dense subset of $[0,m+1]$, and whether almost every $q$ in that interval is such a base is left open.
• § 1, Accumulation points of the set $Y$; proof of Theorem I (pp. 60--72, page images, statements checked, proofs for structure). Lemma 1.1: for Pisot $q$, 0 is not an accumulation point of $Y$ for any $m$ (the distances $d(q^n)$ to the nearest integer decay exponentially, so $|y|>(q+1)^{-1}q^{1-N}$ for every nonzero $y\in Y$). Lemma 1.2: a finite accumulation point of $Y^{q,m}$ makes 0 an accumulation point of $Y^{q,2m}$. Part (a) of Theorem I follows. Lemma 1.3: if $Y$ has no finite accumulation point and $m\ge q-1$, then $q$ is an algebraic integer. Lemmas 1.4 and 1.5, which the paper presents as generalizing results of Frougny [8] and of Erdős, Joó and Schnitzer [6]: if $Y$ has no finite accumulation point and $q$ is algebraic, then for a conjugate $p$ of $q$ a series $\sum s_iq^{-i}=0$ with integers $|s_i|\le m$ also vanishes at $p$ when $|p|>1$ and has partial sums on finitely many circles centered at 0 when $|p|=1$ (Lemma 1.4); for $m\ge q-q^{-1}$ and a complex $p\ne q$ there is such a series vanishing at $q$ but not at $p$ when $|p|>1$, and one whose partial sums at $p$ take infinitely many absolute values when $|p|=1$ (Lemma 1.5). Part (b) of Theorem I follows from Lemmas 1.3--1.5 (p. 64). Lemmas 1.6--1.8 (pp. 65--69): a "lazy" digit algorithm for subseries of a convergent positive series, signed series $\sum s_ia_i=0$ with prescribed sign pattern, and the case $a_i=q^{-[i/m]}$; the proof of Lemma 1.5 (pp. 69--72) treats $p$ nonnegative real, negative real and non-real in turn.
• § 2, Study of $\liminf(y_{k+1}-y_k)$; proof of Theorem II (pp. 72--74). $Y$ is the set of differences $y_k-y_l$, so $\liminf(y_{k+1}-y_k)>0$ iff 0 is not an accumulation point of $Y$, and Part (a) follows from Theorem I. Lemma 2.1 (p. 72, quoted): "(a) If $m\ge q-1$, then $y_{k+1}-y_k\le1$ for every $k$. (b) If $m\le q-1$, then $y_{k+1}-y_k\ge1$ for every $k$. Furthermore, $y_{k+1}-y_k=1$ for infinitely many $k$ and hence $\liminf(y_{k+1}-y_k)=1$." Lemma 2.2 (p. 73): if the set of all differences $u_k-u_l$ of one increasing sequence has a finite accumulation point and another, $(v_k)$, tends to infinity with gaps bounded above, then an increasing sequence containing all the sums $u_k+v_l$ has $\liminf$ of its gaps equal to 0. Part (b) of Theorem II (p. 74) applies Lemma 2.2 with $u=y^{q,n}$, $n=[q-q^{-1}]'$, and $v=y^{q,p}$, $p=[q-1]'$. Proposition 2.3 (p. 74): for $m\le(q-1)/2$, $Y^{q,m}$ has no finite accumulation point.
• § 3, Study of $\limsup(y_{k+1}-y_k)$; proof of Theorems III and IV (pp. 74--80). For a sequence $A=(A_i)$ of nonnegative integers, $y_k^{q,A}$ is the increasing sequence of the sums with $\varepsilon_i\in\{0,1,\ldots,A_i\}$; sums of an $A$-term and a $B$-term lie in the $(A+B)$-sequence. Lemma 3.1 (p. 75, quoted): "If $B$ is periodical with period $d$ and if $q^d\le B_a+1$ for some index $a$, then $y_{k+1}^{q,B}-y_k^{q,B}\le q^a$ for all $k$." Lemma 3.2 (pp. 75--76, "the following important" lemma): for periodic $A,B,C,D$ with (3.1) $Y^{q,A}=\{y_l^{q,A}-y_k^{q,A}\}$ having a finite accumulation point, (3.2) and (3.3) the gaps of $y^{q,B}$ and $y^{q,C}$ bounded, and (3.4) $A+B+C\le D$, the gaps $y_{k+1}^{q,D}-y_k^{q,D}\to0$. Its proof (pp. 76--77) has three steps: from Lemma 2.2, a subsequence $(z_i)$ of $y^{q,A+B}$ with pairwise disjoint power supports and paired differences $z_{2i}-z_{2i-1}$ in $[\delta',q^M\delta')$; from it, finite chains $w_0<\cdots<w_j$ in $y^{q,A+B}$ with steps below $\delta$ and total length above any $\Delta$; then, with the bounded gaps of $y^{q,C}$, every interval of length $\delta$ far enough out meets $y^{q,A+B+C}$, hence $y^{q,D}$. Proof of Theorem III (p. 77): Lemma 3.2 with the constant sequences $A\equiv[q-q^{-1}]'$, $B=C\equiv[q-1]'$, $D\equiv m$, (3.1) by Part (b) of Theorem I and (3.2)--(3.3) by Lemma 2.1 (a). Proof of Theorem IV (pp. 77--78): see theorem_iv. Proposition 3.3 (pp. 78--79): for $m\le q-q^{-1}$, $y_{k+1}-y_k=1$ for infinitely many $k$, so $\limsup(y_{k+1}-y_k)\ge1$ (the open unit intervals $I_j=(m(1+q^2+\cdots+q^{2j})-1,\,m(1+q^2+\cdots+q^{2j}))$ contain no $y_k$). Proposition 3.4 (p. 80): for $q=\sqrt{m+1}$ with $m+1$ not a square, some $c>0$ has every interval $(x-cx^{-1},x+cx^{-1})$ with $x$ large meeting $y^{q,m}$, by the quadratic-irrational approximation theorem cited to Khintchine [9], Theorem 26.
• § 4, Universal developments; proof of Theorem V (pp. 80--82). Lemma 4.1: if $y_{k+1}-y_k\to0$ and $0<\alpha\le1$, every finite block $\delta_1,\ldots,\delta_N$ of digits extends to the left to a finite digit string whose value falls within $q^{-n-N}$ below $\alpha$. The proof of Theorem V (pp. 81--82) applies it in turn to an enumeration $B_1,B_2,\ldots$ of all finite variations, with the rescaled remainders $\alpha_1,\alpha_2,\ldots$ in $(0,1)$, and lets the construction run to infinity.
• References (p. 83), nine items, listed above.

Compiled scope

The paper is compiled at statement depth for its five main results, each on its own page: the result Problem 1096 consumes, Theorem IV (p. 59) with its remarks and its proof (pp. 77--78), read on the page images and paged on theorem_iv; and Theorems I, II, III and V, paged on theorem_i, theorem_ii, theorem_iii and theorem_v, their statements and remarks checked clause by clause on the page images, the reductions in the proofs of Theorems II and III to their lemmas followed, and the proofs of Theorems I and V read for structure only. The lemmas and propositions are recorded here as statements read on the page images; their proofs were read for structure only. A filing observation, not a review verdict, is recorded on the Theorem IV result page: the proof of Theorem IV opens its first case with the strict inequality $q<2^{1/4}$ where the theorem allows equality, and the argument applies unchanged at $q=2^{1/4}$. On notation, the paper's $p_1,p_2$ (the first and second Pisot numbers, $\approx1.325$ and $\approx1.380$) are the site's $q_0,q_1$ on the problem page. Nothing here is independently reviewed.

Bears on. #1096: Theorem IV (printed p. 59, PDF p. 3), the gaps $y_{k+1}-y_k$ tending to 0 for every $m\ge1$ when $1<q\le2^{1/4}$ and $q$ is not the square root of the second Pisot number, with $m=1$ is the problem's sequence and question: the introduction (p. 57) names the question as [4], Problem 4, the 1990 Bulletin paper's Problem 4, and says "One of the purposes of this paper is to give an affirmative answer to this question", announcing the $m=1$ case there for every $q$ between 1 and $2^{1/4}$ other than $\sqrt{p_2}$, whose value it prints as $\approx1.175$. Every $q$ in $(1,\sqrt{p_2})$ is covered, so the problem's $\epsilon$ may be any number at most $\sqrt{p_2}-1\approx0.175$, and every $q$ in $(\sqrt{p_2},2^{1/4}]$ as well. Of the two second-hand accounts the problem page recorded, Feng's ($1<q\le2^{1/4}$ with the possible exception of $\sqrt{p_2}$) is the printed statement, and the site's ($1<q<\sqrt{q_1}\approx1.175$) is its part below the excluded point. The proof (pp. 77--78) covers $q=\sqrt{p_1}\approx1.151$, where Feng's Theorem 1.4 is silent, by a separate case through $q^3$. The paper's Theorem I (b) (p. 58), applied with $m=1$ at the base $q^2$ (at $q^3$ for $q=\sqrt{p_1}$), supplies the first hypothesis of Lemma 3.2 in that proof (p. 78); Theorem I and Lemma 3.2 were read for structure only. The other main results are context for the problem, not resolutions of it: Theorem II (a) (p. 59) gives $\liminf(y_{k+1}-y_k)>0$ for every $m$, hence for $m=1$, at every Pisot $q$, and every Pisot number is at least $q_0\approx1.3247$, above the range of Theorem IV, while Part (b) never applies with $m=1$; Theorem III (p. 59) never applies for $m=1$ or $m=2$, as the paper says; and Theorem V (p. 60) takes the gaps tending to 0 as its hypothesis and draws universal developments from it.

Results.

• Theorem IV (p. 59): for $1<q\le2^{1/4}$ and $q\ne\sqrt{p_2}$, $y_{k+1}-y_k\to0$ for every $m\ge1$; proved on pp. 77--78 from Theorem I (b) and Lemmas 3.1 and 3.2.
• Theorem I (p. 58): $Y^{q,m}$ has no finite accumulation point for any $m$ when $q$ is Pisot, and has one for every integer $m\ge q-q^{-1}$ when $q$ is not; proved in § 1 (pp. 60--72).
• Theorem II (p. 59): $\liminf(y_{k+1}-y_k)>0$ for every $m$ when $q$ is Pisot, and $=0$ for every $m\ge[q-q^{-1}]'+[q-1]'$ when $q$ is not; proved in § 2 (pp. 72--74).
• Theorem III (p. 59): for non-Pisot $q$, $y_{k+1}-y_k\to0$ for every $m\ge[q-q^{-1}]'+2[q-1]'$; proved on p. 77 from Lemma 3.2.
• Theorem V (p. 60): if $y_{k+1}-y_k\to0$ then every $0<\alpha<m/(q-1)$ has a universal development; proved in § 4 (pp. 80--82).

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