Feng 2016 topology polynomials bounded integer coefficients

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corollary_1_3: Feng's corollary that the lower limit of the consecutive gaps of the ordered sums of powers of q with digits 0, ..., m is zero exactly when q < m+1 and q is not a Pisot number.

corollary_1_7: Feng's answer to Lau's question: every q in (1, 2) for which only finitely many values at q of polynomials with coefficients ±1 and 0 lie in [-1/(q-1), 1/(q-1)] is a Pisot number.

theorem_1_11: Feng's theorem that a homogeneous iterated function system x -> rho x + b_i on the line, with 0 = b_0 < ... < b_m = 1 - rho and consecutive gaps b_{i+1} - b_i at most rho, satisfies the finite type condition whenever it satisfies the weak separation condition.

theorem_1_2: Feng's main theorem: for q > 1 and a positive integer m, the set Y_m(q) of values at q of polynomials with coefficients in {0, ±1, ..., ±m} is dense in the reals exactly when q < m+1 and q is not a Pisot number.

theorem_1_4: Feng's 2016 theorem that for 1 < q < sqrt(m+1) with q^2 not a Pisot number the upper limit L_m(q) of the consecutive gaps of the finite sums of powers of q with digits 0, ..., m is zero; with m = 1 and the absence of Pisot numbers below q_0 = 1.3247..., this gives x_{k+1} - x_k -> 0 for every q in (1, sqrt(q_0)), the question of Problem 1096.

theorem_1_6: Feng's theorem, conjectured by Akiyama and Komornik, that for 1 < q <= m+1 the set Y_m(q) of values at q of polynomials with coefficients in {0, ±1, ..., ±m} has no finite accumulation point exactly when 0 is not one of its accumulation points.

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Feng, De-Jun, On the topology of polynomials with bounded integer coefficients. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 1, 181--193; DOI 10.4171/JEMS/587 (the Crossref record dates the article 16 December 2015). The site's key Fe16.

The copy read for this card is arXiv:1109.1407v3 (1 February 2015; "15 pages, to appear in J. Eur. Math. Soc"), 15 pages with a complete text layer; the abstract page lists three versions (v1 7 September 2011, v2 10 November 2011, v3 1 February 2015) and no journal reference. The journal text was not compared; the locators below are the preprint's. The statements were read on the rendered page images of pp. 1--6. Source: https://arxiv.org/abs/1109.1407. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1109.1407), every other right reserved.

Read status: claims checked for the abstract, Question 1.1, the two non-density cases with their proofs, Theorem 1.2, the definitions of $X_m(q)$, $\ell_m(q)$ and $L_m(q)$, Corollary 1.3, the survey paragraph on partial results, Theorem 1.4 with its footnote, Theorems 1.5 and 1.6, read clause by clause on the page images of pp. 1--3; Corollary 1.7 and the definition of an F-number (p. 4), Definitions 1.8 and 1.9, Remark 1.10, Theorem 1.11 and the derivation of Theorem 1.6 from it (pp. 4--6), read clause by clause on the page images; the proof of Theorem 1.11 (Section 2, pp. 6--11) was read for structure only and not checked.

For q > 1 and a positive integer m let Y_m(q) be the set of values sum_{i=0}^n eps_i q^i with digits eps_i in {0, ±1, ..., ±m}. The paper's main theorem is that Y_m(q) is dense in R exactly when q < m+1 and q is not Pisot, completing a chain of partial results and answering an open question of Erdős, Joó and Komornik. The two known non-density cases are recalled with short proofs: if q is Pisot, multiplying a nonzero polynomial value by its conjugates gives a nonzero integer and hence |P(q)| > m^{-d}(1-rho)^d, so 0 is isolated and (since Y_{2m}(q) = Y_m(q) - Y_m(q)) Y_m(q) is uniformly discrete (Garsia); if q

= m+1 the digit sums cannot bridge the gap below q^n (Erdős-Komornik). The positive direction, that density holds for all non-Pisot q < m+1, is proved by iterated function system techniques, the paper's stated keywords being Pisot numbers and iteration function systems. For problem 1096 the operative statement is not the main theorem but Theorem 1.4 (p. 3): for $1<q<\sqrt{m+1}$ with $q^2$ not a Pisot number, $L_m(q)=0$, and in particular $L_1(q)=0$ for $q\in(1,\sqrt2)$ with $q^2$ not Pisot, where $L_1(q)$ is the upper limit of the gaps $x_{n+1}-x_n$ of the problem's sequence; since no Pisot number lies in $(1,q_0)$, $q_0\approx1.3247$ the smallest Pisot number, the gaps tend to $0$ for every $1<q<\sqrt{q_0}$, which answers the problem (the deduction is written on the problem page).

Contents

• Section 1 (pp. 1--6). Question 1.1: for which $(q,m)$ is $Y_m(q)$ dense in $\mathbb R$? The non-density cases (Garsia for Pisot $q$; Erdős--Komornik for $q\ge m+1$) with their proofs (pp. 1--2). Theorem 1.2 (p. 2): "$Y_m(q)$ is dense in $\mathbb R$ if and only if $q<m+1$ and $q$ is not a Pisot number." The Erdős--Joó--Komornik project: $X_m(q)=\{\sum_{i=0}^n\varepsilon_iq^i:\varepsilon_i\in\{0,1,\ldots,m\}\}$ arranged as $0=x_0(q,m)<x_1(q,m)<\cdots$, with $\ell_m(q)=\liminf(x_{n+1}(q,m)-x_n(q,m))$ and $L_m(q)=\limsup(x_{n+1}(q,m)-x_n(q,m))$; by Drobot, $\ell_m(q)=0$ iff $Y_m(q)$ is dense; Corollary 1.3: $\ell_m(q)=0$ iff $q<m+1$ and $q$ is not Pisot, answering the question of [8] whether $\ell_1(q)=0$ for every non-Pisot $q\in(1,2)$. The survey paragraph (p. 3): Bugeaud (some $m$ with $\ell_m(q)=0$ for non-Pisot $q$), Erdős--Komornik [9] ($\ell_m(q)=0$ for non-Pisot $q$ and $m\ge\lceil q-q^{-1}\rceil+\lceil q-1\rceil$), Akiyama--Komornik, Sidorov--Solomyak; for $L_m(q)$: Erdős--Komornik proved $L_m(q)>0$ for Pisot $q$ or $q\ge(m+\sqrt{m^2+4})/2$, Komornik's conjecture $L_1(q)=0$ for every non-Pisot $q$ below the golden ratio, "$L_1(q)=0$ if $1<q\le\sqrt[3]2\approx1.2599$ ([9, 1]). Here the second part was only proved in [9] for all $1<q\le\sqrt[4]2\approx1.1892$ with the possible exception of the square root of the second Pisot number." Theorem 1.4 (p. 3): $L_m(q)=0$ whenever $1<q<\sqrt{m+1}$ and $q^2$ is not Pisot, so $L_1(q)=0$ for every $q\in(1,\sqrt2)$ whose square is not Pisot; derived from Corollary 1.3 and the implication $\ell_m(q^2)=0\Rightarrow L_m(q)=0$ ([1, Lemma 2.5]; footnote 3: "first proved in [8, Theorem 5] in the case $m=1$"). Theorem 1.5 (Akiyama--Komornik): $Y_m(q)$ has a finite accumulation point iff $q<m+1$ and $q$ is not Pisot. Theorem 1.6 (the paper's new result): for $1<q\le m+1$, $Y_m(q)$ has no finite accumulation point iff $0$ is not an accumulation point; Theorem 1.2 follows from Theorems 1.5 and 1.6. Corollary 1.7 (every F-number is Pisot); the set $A(q)$ of $\pm1$ sums (p. 4); homogeneous iterated function systems (p. 4), with the weak separation and finite type conditions (Definitions 1.8 and 1.9, p. 5) and Theorem 1.11 (p. 5), from which Theorem 1.6 is derived (pp. 5--6); Corollary 1.12 (p. 6): the IFS $\{q^{-1}x+i(1-q^{-1})/m\}_{i=0}^m$, for $1<q<m+1$, satisfies the weak separation condition (resp. the finite type condition) if and only if $q$ is a Pisot number.
• Section 2 (pp. 6--11): Lemma 2.1 (p. 6) and Lemma 2.2 (p. 8), then the proof of Theorem 1.11 on separation properties of homogeneous IFS on $\mathbb R$ in three steps (pp. 8--11); read for structure only.
• Section 3 (pp. 11--13): final remarks and open questions, among them Proposition 3.1 and Corollary 3.2 (p. 12) and the two closing questions (p. 13), read for structure only; references, pp. 13--15 (not read beyond the headings).

Compiled scope

The introduction (Section 1, pp. 1--6) was read clause by clause on the page images; Section 2 and Section 3 were read for structure only. Result pages:

• Theorem 1.2 (p. 2), the main theorem: $Y_m(q)$ is dense in $\mathbb R$ if and only if $q<m+1$ and $q$ is not a Pisot number.
• Corollary 1.3 (p. 3): $\ell_m(q)=0$ if and only if $q<m+1$ and $q$ is not a Pisot number.
• Theorem 1.4 (p. 3): if $1<q<\sqrt{m+1}$ and $q^2$ is not a Pisot number, then $L_m(q)=0$; in particular $L_1(q)=0$ for $q\in(1,\sqrt2)$ with $q^2$ not Pisot.
• Theorem 1.6 (p. 3), the result the paper proves: for $1<q\le m+1$, $Y_m(q)$ has no finite accumulation points in $\mathbb R$ if and only if $0$ is not an accumulation point of $Y_m(q)$.
• Corollary 1.7 (p. 4): every F-number is a Pisot number.
• Theorem 1.11 (p. 5): a homogeneous IFS satisfying (1.1) and (1.2) that satisfies the weak separation condition satisfies the finite type condition.

The non-density cases (pp. 1--2), Theorem 1.5 (Akiyama and Komornik's, recalled), Corollary 1.12 and Section 3 are recorded in the contents above and have no page of their own. No proof was checked and nothing here is independently reviewed.

Bears on. #1096: Theorem 1.4 with $m=1$ gives $\lim(x_{n+1}-x_n)=0$ for every $q\in(1,\sqrt2)$ whose square is not a Pisot number, hence for every $1<q<\sqrt{q_0}\approx1.151$, since then $q^2\in(1,q_0)$, an interval that contains no Pisot number. Erdős and Komornik's Theorem IV, whose range Feng's survey paragraph reports as $1<q\le2^{1/4}$ with the possible exception of the square root of the second Pisot number $q_1$, covers the longer interval $(1,\sqrt{q_1})$, $\sqrt{q_1}\approx1.175$; Theorem 1.4 itself reaches every $q\in(1,\sqrt2)$ whose square is not Pisot and is silent at the square roots of Pisot numbers. Theorem 1.2 and Corollary 1.3 concern density and the lower limit $\ell_1(q)$ of the gaps, not the problem's limit; Corollary 1.3 at $q^2$ is the input of Theorem 1.4. Theorems 1.6 and 1.11 and Corollary 1.7 bear on no Erdős problem directly.

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