Erdos 1996 pisot numbers

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theorem: The 1996 Erdős-Joó-Schnitzer theorem that for q strictly between 1 and the golden ratio, q is a Pisot number exactly when the finite sums of powers of q with digits 0, 1, 2 have a positive least gap l_2(q); a single-digit-set strengthening of Bugeaud's characterization through all digit sets.

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Erdős, P. and Joó, I. and Schnitzer, F. J., On {P}isot numbers. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 39 (1996), 95--99. "Received October 6, 1995" (p. 95). The site's key EJS96. No Crossref record exists for the paper (bibliographic query, 2026-09-18).

The copy read for this card is not the paper alone: it is the journal archive's file of the whole volume 39 (1996), 182 pages, a re-typeset volume (dvips and Distiller; every page carries the typesetting footer "2016. december 23.") with a text layer in which the mathematical symbols are garbled. The paper occupies printed pp. 95--99, which are PDF pp. 95--99 (physical page equals printed page throughout the file); the Theorem was read on the rendered page image of p. 95. No other page of the volume is cited by this card or by the pages that consume it. Source: https://annalesm.elte.hu/archive.html. No notice is printed in the volume file (the title pages, PDF pp. 1--2, and the last two pages read); the journal's archive page (https://annalesm.elte.hu/archive.html, read 2026-10-02) states no copyright, license or terms; the term is unstated.

Read status: claims checked for the introduction and the Theorem, read clause by clause on the page image of p. 95, and for the proof's closing paragraph, the two open problems and the reference list on the page image of p. 99; the five lemmas (pp. 96--99) are not consumed: their statements and the two remarks on Lemma 4 were read on the page images of pp. 96 and 98 for the digest below, and their proofs were not checked.

For 1<q<2 let Y be the set of finite sums Σ ε_i q^i with integer digits 0 ≤ ε_i ≤ 2, and let l_2(q) be the infimum of |y_1-y_2| over distinct y_1,y_2 ∈ Y (with l_k(q) the analog for |ε_i| ≤ k). Bugeaud had shown that q is Pisot if and only if l_k(q)>0 for all k ≥ 1, and earlier work gave the single implication q Pisot ⇒ l_k(q)>0. The paper's Theorem strengthens this to a single value of k in a restricted range: for 1<q<A=(1+√5)/2, q is a Pisot number if and only if l_2(q)>0. The proof runs through a chain of lemmas: l_2(q)>0 forces only finitely many distances at most 1/(q-1) among sums with digits 0,1 (Lemma 1), hence q is an algebraic integer (Lemma 2); Lemma 3 transfers vanishing power series Σ s_n q^{-n}=0 with digits 0,±1 to the conjugates with |q_i|>1; Lemma 4 constructs, for q ≤ A, a nontrivial expansion with digits 0,±1 splitting the naturals into two half-plane classes determined by the arguments of the conjugate's powers, which with Lemma 3 rules out conjugates of modulus above one (the remark after Lemma 4); Lemma 5 rules out conjugates of modulus one. The relevant part for problem 1096 is this spacing function: the problem asks whether, for q slightly above 1, the gaps x_{k+1}-x_k in the ordered set of sums of distinct powers of q tend to 0, and the theorem shows that a positive lower bound on such spacings (in the digit-2 form) is equivalent to q being Pisot, which fails for q close to 1.

Contents

• P. 95: the definitions ($Y$ with digits $0\le\varepsilon_i\le2$; $l_2(q)=\inf\{|y_1-y_2|:y_1,y_2\in Y,\ y_1\ne y_2\}$; $l_k(q)$ with $|\varepsilon_i|\le k$); the recall that Bugeaud proved, for $1<q<2$, "$q$ is Pisot $\Leftrightarrow$ $l_k(q)>0$ for all $k\ge1$" ([2]) and that a former result ([8]) gives $q$ Pisot $\Rightarrow l_1(q)>0$, whose proof shows $q$ Pisot $\Rightarrow l_k(q)>0$ for all $k$; the Theorem: "Let $1<q<A=\frac{1+\sqrt5}2$. Then $q$ is Pisot $\Leftrightarrow l_2(q)>0$"; Remark: "The proof improves some ideas from [7]."
• Pp. 96--99: Lemmas 1--4 (p. 96) and Lemma 5 (p. 98) with their proofs, the proof of the Theorem and two open problems (p. 99).

Compiled scope

Pages 95 and 99 were read on the page images, and the lemma statements and the two remarks on Lemma 4 on pp. 96 and 98; the Theorem is compiled as a statement with a proof pointer; no proof was checked and nothing here is independently reviewed.

Bears on. #1096, as the site's source for the digit-$2$ characterization: for $1<q<(1+\sqrt5)/2$, the spacings of the sums with digits $0,1,2$ stay bounded away from $0$ exactly when $q$ is Pisot; a result about a denser sequence than the problem's, which the site records as the improvement of Bugeaud's characterization, and which does not decide the problem: for Pisot $q$ it bounds the gaps of the problem's sums (a subset of the digit-$2$ sums) away from $0$, but the smallest Pisot number is about $1.3247$, and for non-Pisot $q$ its small gaps may use the digit $2$.

Results to transcribe.

• Theorem (p. 95): For 1<q<A=(1+√5)/2, q is a Pisot number if and only if l_2(q)>0, where l_2(q) is the least gap between distinct sums Σ ε_i q^i with digits 0 ≤ ε_i ≤ 2.
• Lemma 1: l_2(q)>0 implies that only finitely many distances at most 1/(q-1) occur between numbers of the form Σ ε_i q^i with ε_i ∈ {0,1}.
• Lemma 2: l_2(q)>0 implies that q is an algebraic integer, by pigeonholing the finitely many values of q^n - Σ ε_i q^i.
• Lemma 3: If l_2(q)>0 then a vanishing expansion Σ s_n q^{-n}=0 with digits s_n ∈ {0,±1} forces Σ s_n q_i^{-n}=0 for every conjugate with |q_i|>1.
• Lemma 4 (p. 96): For l_2(q)>0 and q ≤ A, and angles α ≠ 0 and β with |α|, |β| ≤ π such that nα is never parallel to β, split the naturals into N_1 and N_2 by the side of the line of angle β on which the ray of angle nα lies; then some nontrivial digits ε_n ∈ {0,1} give equal sums of ε_n q^{-n} over N_1 and over N_2. The remark after the statement combines this with Lemma 3 to exclude a conjugate of modulus above one; a second remark (p. 98) says the lemma fails for q > A.
• Lemma 5: For l_2(q)>0 and q ≤ A, q has no conjugate of modulus one.

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