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Statement

Setting (p. 95): "The algebraic integer $1<q<2$ is called a Pisot number if $|q_i|<1$ for all of its conjugates"; $Y=\{\sum_0^n\varepsilon_iq^i: n\ge0,\ \varepsilon_i\in\mathbb Z,\ 0\le\varepsilon_i\le2\}$ and $l_2(q)=\inf\{|y_1-y_2|: y_1,y_2\in Y,\ y_1\ne y_2\}$; analogously $l_k(q)$ with $|\varepsilon_i|\le k$ in place of $|\varepsilon_i|\le2$ (so printed, although $Y$ is written with $0\le\varepsilon_i\le2$; the page does not say whether $l_k$ takes the digits $0,\ldots,k$ or $-k,\ldots,k$). As printed:

Theorem. Let $1<q<A=\frac{1+\sqrt5}2$. Then

$$q\ \text{is Pisot}\ \Leftrightarrow\ l_2(q)>0.$$

The introduction (p. 95) records the results it strengthens: Bugeaud proved ([2]) that for $1<q<2$, $q$ is Pisot if and only if $l_k(q)>0$ for all $k\ge1$; a former result ([8]) gives $q$ Pisot $\Rightarrow l_1(q)>0$, and "The same proof shows that $q$ is Pisot $\Rightarrow l_k(q)>0$ for all $k$." The Remark: "The proof improves some ideas from [7]." The site's Problem 1096 page states the theorem with $\liminf(x^2_{k+1}-x^2_k)>0$ in place of $l_2(q)>0$; the two agree because for this sequence the infimum of the gaps equals their lower limit (the observation of Erdős, Joó and Komornik's 1998 Acta Arithmetica paper, p. 201, for the digit set $\{0,1\}$; the same argument, adding a large power of $q$ to both ends of a small gap, applies to digits $\{0,1,2\}$ -- an authored remark, not checked further here).

Source. P. Erdős, I. Joó and F. J. Schnitzer, On Pisot numbers, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 39 (1996), 95--99; the Theorem on printed p. 95, which is PDF p. 95 of the volume file, read on the rendered page image. The edition is identified in the source digest.

Read depth. Claims checked: the definitions, the recalled results and the Theorem were read clause by clause on the page image of p. 95. The statements of Lemmas 1--5 (pp. 96 and 98), the two remarks on Lemma 4 (pp. 96 and 98) and the closing paragraph (p. 99) were read on the page images for the proof pointer below; the proofs of the lemmas were not checked.

Proof pointer

Pp. 96--99, through five lemmas: $l_2(q)>0$ leaves only finitely many distances at most $1/(q-1)$ among the sums with digits $0,1$ (Lemma 1), so $q$ is an algebraic integer (Lemma 2); a vanishing series $\sum s_nq^{-n}=0$ with digits $0,\pm1$ transfers to every conjugate of modulus above $1$ (Lemma 3); for $q\le A$ a half-plane partition of the exponents by the arguments of a conjugate's powers produces a nontrivial expansion with digits $0,\pm1$ (Lemma 4), which with Lemma 3 excludes conjugates of modulus above $1$ (the remark after the statement of Lemma 4, p. 96); Lemma 5 excludes conjugates of modulus $1$. A second remark, after the proof of Lemma 4 (p. 98), says that Lemma 4 fails for $q>A$: no nontrivial expansion splits the odd and the even exponents, since $1/q+1/q^3+\cdots<1$ there. The closing paragraph (p. 99) derives that $q$ is Pisot from $l_2(q)>0$ by the lemmas and takes the converse, $l_k(q)>0$ for every $k$ when $q$ is Pisot, from [2], whose proof it calls identical to that of [3] "for the special case $k=l$" (so printed, evidently for $k=1$). Two open problems follow: whether the Theorem holds for $A<q<2$, and whether $q$ is Pisot $\Leftrightarrow l_1(q)>0$. Not reconstructed here.

Dependencies

The proofs of Lemmas 3 and 5 rest on the proof of Corollary 3.2 of the paper's [5] (C. Frougny, Representations of numbers and finite automata, Math. Systems Theory 25 (1992), 37--60; not held): Lemma 3's proof is declared identical to it, and Lemma 5's proof reads off from it that the partial sums at a conjugate lie in one of finitely many circles centred at the origin. For the converse direction, the paper's [2] (Y. Bugeaud, On a property of Pisot numbers and related questions, Acta Math. Hungar. 73 (1996), 33--39, cited as "to appear"; not held) and [3] (Bogmér, Horváth and Sövegjártó, Acta Math. Hungar. 58 (1991), 153--155; not held); the Remark's [7] is Berend and Frougny, Math. Systems Theory 27 (1994), 275--282, and [8] is Erdős, Joó and Komornik's Acta Arithmetica paper, cited as an IRMA preprint (held).

Bears on

• Problem 1096: the site's digit-$2$ characterization, "if $1<q<(1+\sqrt5)/2$, then $q$ is a Pisot--Vijayaraghavan number if and only if $\liminf(x^2_{k+1}-x^2_k)>0$"; context for the problem's digit-$\{0,1\}$ sequence, whose gaps it bounds away from $0$ at Pisot $q$ below $(1+\sqrt5)/2$, all above $1.32$, while at other $q$ it says nothing about them, so it does not decide the problem.