Problem 1096

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claims/: The 4 claim pages of Problem 1096, one per claimant's result; the problem's standing derives from them.

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Statement. Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$.

Is it true that, provided $\epsilon>0$ is sufficiently small, $x_{k+1}-x_k \to 0$?

Formulation. The site's wording (page last edited 16 April 2026). The sums run over finite sets $S$ of nonnegative integers, the empty sum giving $x_1=0$; the sequence begins $0,1,q$ (the site's remark; the 1998 sources write $y_0=0$, $y_1=1$, $y_2=q$, while [EJK90] writes $0=:y_1$, so $y_2=1$, $y_3=q$, p. 386). The question asks for one $\epsilon>0$ such that for every $q$ in $(1,1+\epsilon)$ the consecutive gaps tend to $0$. In the sources the sequence is $(y_n)$ with $l(q)=\inf(y_{k+1}-y_k)$, which equals $\liminf(y_{k+1}-y_k)$, and $L(q)=\limsup(y_{k+1}-y_k)$ (Erdős, Joó and Komornik 1998, p. 201), or $X_1(q)$ with $\ell_1(q)$ and $L_1(q)$ (Feng), so the question is whether $L(q)=0$ on some interval $(1,1+\epsilon)$. The site's sources are [EJK90] and [GWNT91]; the earliest located statement of the question is Problem 4 of the 1990 Bulletin paper (p. 389): "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 1991 problem session the site names is problem 91:18 of the 1991 Western Number Theory problem set (p. 16), which asks for a proof that $x_{k+1}-x_k\to0$ when $\epsilon$ is small and guesses that every $q<q_0$, $q_0^3=q_0+1$, has this property. The site's label PROVED (LEAN) carries a catalog suffix explained under Formalization.

Status. The site's label is PROVED (LEAN), whose catalog suffix is explained under Formalization. The first resolution is Theorem IV of Erdős and Komornik's 1998 paper (Acta Math. Hungar. 79 (1998), 57--83, refereed), an accepted full claim on its page: "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 $m\ge1$", where $(y_k)$ is the ordered sequence of the sums with digits $0,\ldots,m$ and $m=1$ gives the problem's sequence; its introduction (p. 57) names the question as [EJK90]'s Problem 4 and says "One of the purposes of this paper is to give an affirmative answer to this question". With $2^{1/4}\approx1.1892$ and $\sqrt{q_1}\approx1.1749$ ($q_1\approx1.38$, the second Pisot number, the paper's $p_2$), every $q$ in $(1,\sqrt{q_1})$ is covered, so the answer is yes, with $\epsilon=\sqrt{q_1}-1\approx0.175$, and so is every $q$ in $(\sqrt{q_1},2^{1/4}]$. Theorem 1.4 (i) of Akiyama and Komornik (J. Number Theory 133 (2013), 375--390, refereed; quoted from the arXiv text), the second accepted full claim, on their page, gives $L_1(q)=0$ for every $1<q\le2^{1/3}\approx1.2599$, a range that contains the excluded point $\sqrt{q_1}$ and so closes it. A third first-hand source, the third accepted full claim, on Feng's page, is Theorem 1.4 of Feng's paper (J. Eur. Math. Soc. 18 (2016), 181--193, refereed; quoted from arXiv v3): for $1<q<\sqrt2$ with $q^2$ not a Pisot number, $\lim(x_{n+1}-x_n)=0$. Since $q_0\approx1.3247$, the real root of $x^3=x+1$, is the smallest Pisot number (Siegel's classical theorem, which the site's commentary also states), every $q$ in $(1,\sqrt{q_0})$ has $q^2\in(1,q_0)$ not Pisot and $q<\sqrt2$, so the gaps tend to $0$ for every $1<q<\sqrt{q_0}\approx1.1510$, with $\epsilon=\sqrt{q_0}-1\approx0.151$ (an authored deduction, one line, from Feng's theorem and Siegel's theorem). The site reports the 1998 range as $1<q<\sqrt{q_1}$ and Feng's introduction as $1<q\le2^{1/4}$ "with the possible exception of the square root of the second Pisot number"; the printed theorem is Feng's version, and the site's range is its part below the excluded point. The frontmatter standing is derived from the three accepted pages; a fourth, pending claim, the thread's two-page note on Acosta De León's page, repeats the deduction from Feng's theorem and adds nothing to the standing.

Source. erdosproblems.com/1096, accessed 2026-09-18: the problem page (PROVED (LEAN), with the site's note that the answer is affirmative and the proof verified in Lean; last edited 16 April 2026; source keys [EJK90], [GWNT91]; commentary citing [Bu96], [EJS96], [ErKo98] and [Fe16]; the formalized-statement indicator answering yes; the page thanks van Doorn and one other contributor), its three-comment discussion thread (16 April 2026) and its empty proof-claims tab. Cite as: T. F. Bloom, Erdős Problem #1096, https://www.erdosproblems.com/1096, accessed 2026-09-18.

References.

• [EJK90] Erdős, Pál and Joó, István and Komornik, Vilmos, Characterization of the unique expansions $1=\sum^\infty_{i=1}q^{-n_i}$ and related problems. Bull. Soc. Math. France 118 (1990), no. 3, 377--390, doi:10.24033/bsmf.2151 (Crossref record accessed). Theorem 4, p. 386; Problem 4, p. 389. Library home: erdos_1990_characterization_unique_expansions_related_problems.
• [GWNT91] The site's key for "the 1991 problem session of Great Western Number Theory". Located: problem 91:18 (Paul Erdős & I. Joó), p. 16, of Western Number Theory Problems, 1991-12-19 & 22, edited by Richard K. Guy; not held. Library home: guy_1991_western_number_theory_problems.
• [Bu96] Bugeaud, Y., On a property of Pisot numbers and related questions. Acta Math. Hungar. 73 (1996), no. 1--2, 33--39, doi:10.1007/BF00058941 (Crossref bibliographic query). Not held (publisher paywall). Its theorem is quoted from the site and from [EJS96], p. 95: for $1<q<2$, $q$ is Pisot if and only if $l_k(q)>0$ for all $k\ge1$.
• [EJS96] Erdős, P. and Joó, I. and Schnitzer, F. J., On Pisot numbers. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 39 (1996), 95--99 (received 6 October 1995; no Crossref record). The Theorem, p. 95, in the journal archive's volume file. Library home: erdos_1996_pisot_numbers.
• [ErKo98] Erdős, P. and Komornik, V., Developments in non-integer bases. Acta Math. Hungar. 79 (1998), no. 1--2, 57--83, doi:10.1023/A:1006557705401 (Crossref record accessed; received 30 September 1996, per p. 83). Abstract and introduction, p. 57; Theorem IV with its remarks, pp. 59--60; its proof, pp. 77--78. Feng's reference [9]. Library home: erdos_komornik_1998_developments_non_integer_bases.
• [Fe16] 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 (Crossref record accessed); arXiv:1109.1407v3 (1 February 2015, "to appear in J. Eur. Math. Soc"; the journal text not compared). Theorem 1.2, p. 2; Corollary 1.3 and Theorem 1.4, p. 3. Library home: feng_2016_topology_polynomials_bounded_integer_coefficients.
• [EJK98] Erdős, P., Joó, I. and Komornik, V., On the sequence of numbers of the form $\varepsilon_0+\varepsilon_1q+\ldots+\varepsilon_nq^n$, $\varepsilon_i\in\{0,1\}$. Acta Arith. 83 (1998), no. 3, 201--210, doi:10.4064/aa-83-3-201-210 (Crossref record accessed). Not a site key. Theorem 4, p. 206; Theorem 5, p. 207. Library home: erdos_1998_sequence_numbers_form_sums_powers_q.
• [Si44] Siegel, C. L., Algebraic integers whose conjugates lie in the unit circle. Duke Math. J. 11 (1944), 597--602. The classical source of the fact that $q_0$ is the smallest Pisot number; not held, cited as the standard reference for a fact the site's commentary also states.

Formalization. The suffix of the site's label PROVED (LEAN) is a catalog label. The file ErdosProblems/1096.lean of formal-conjectures at the linked commit declares erdos_1096 : answer(True) ↔ ∃ ε > 0, ∀ q, 1 < q → q < 1 + ε → ∀ x : ℕ → ℝ, StrictMono x → Set.range x = { ∑ i ∈ S, q ^ i | S : Finset ℕ } → Tendsto (fun k => x (k + 1) - x k) atTop (𝓝 0) under category research solved, AMS 11, with proof sorry, a docstring attributing the solution to Erdős and Komornik for $1<q<\sqrt{q_1}$, and a formal_proof attribute naming src/latest/ErdosProblems/Erdos1096.lean in the repository plby/lean-proofs at its commit of 30 August 2026, linked at that commit from Erdős and Komornik's claim page. That external file at that commit (2,645 bytes, 85 lines) is headed leanprover/lean4:v4.33.0 mathlib v4.33.0, imports a companion module ErdosProblems.Erdos1096.Erdos1096Accumulation of the same repository (not examined), names Erdős and Komornik as informal authors and Codex and GPT-5.6 Sol as formal authors, and proves at line 44 theorem erdos_1096 with the right-hand side of the collection's statement, taking $\varepsilon=1/1000$ and deriving the conclusion from three lemmas of the imported module (small differences in the spectrum of $q^2$ for $q^2<1.01$, eventual right-density of the spectrum of $q$, and gaps tending to zero from that density), the route of the Erdős--Joó--Komornik and Feng arguments through $q^2$; the file contains no sorry and no axiom declaration, and its #print axioms erdos_1096 (line 81) has no recorded output; its comment says the detailed proof is in a TeX file of the repository (not examined). The imported module is not examined, this corpus has not built or audited the development, and no kernel credit is claimed. The community database (teorth/erdosproblems) records proved (Lean), as of its last update on 23 August 2026, the statement formalized, as of its last update on 21 May 2026, formal_status Lean and no formal-proof URL; the site's indicator answers yes.

Current assessment

The question (site formulation, accessed 2026-09-18). The statement above; PROVED (LEAN); last edited 16 April 2026. The site's commentary attributes the problem to Erdős and Joó at the 1991 problem session of the Western Number Theory conference, where, it says, they guessed the threshold to be $q_0\approx1.3247$, the real root of $x^3=x+1$ and the smallest Pisot number; it credits [EJK90] with excluding every Pisot number and with the bound $x_{k+1}-x_k\le1$ for every $k$ when $1<q\le2$, notes that the sequence starts $0,1,q$, and then states Bugeaud's characterization ($1<q\le2$ is Pisot iff $\liminf(x^m_{k+1}-x^m_k)>0$ for all $m\ge1$, where $x^m_k$ runs over the sums with digits in $\{0,\ldots,m\}$), the Erdős--Joó--Schnitzer improvement (for $1<q<(1+\sqrt5)/2$, Pisot iff $\liminf(x^2_{k+1}-x^2_k)>0$), the first resolution by Erdős and Komornik [ErKo98], with $\lim(x_{n+1}-x_n)=0$ for $1<q<\sqrt{q_1}\approx1.175$ where $q_1\approx1.38$ is the second Pisot number, and Feng's two results ($\liminf(x_{n+1}-x_n)=0$ iff $1<q<2$ is not Pisot; if $1<q<\sqrt2$ and $q^2$ is not Pisot then $\lim(x_{n+1}-x_n)=0$). The thread (16 April 2026): a first comment links a proof, described by its poster as unverified, hosted on a data repository (a two-page note dated 16 April 2026; see Beyond the question); the curator replies that Feng's paper solves the problem, through its Theorem 1.4 and the gap between $1$ and the smallest Pisot number, a consequence Feng does not state, and amends the reply the same day after finding in Feng's paper that Erdős and Komornik had already solved the question in 1998, updating the site; a third comment notes that Feng's paper had been reported on the forum's missing-problems thread. The proof-claims tab is empty. The community database record says proved (Lean), as of its last update on 23 August 2026.

The origin and the early results. Problem 4 of [EJK90] (p. 389) is quoted in the Formulation paragraph; the same paper's Theorem 4 (p. 386) gives a) $y_{n+1}-y_n\le1$ for all $n\ge1$ (the bound the site's commentary states), b) $y_{n+1}-y_n=1$ infinitely often for $q>(1+\sqrt5)/2$, c) if $y_{n+1}-y_n\to0$ then $1$ has an infinite expansion with arbitrarily long runs of zero digits, and d) some Pisot $q<(1+\sqrt5)/2$ (the real root of $q^3=q^2+1$) has $y_{n+1}-y_n\not\to0$, through c) and the fact, recalled there from two papers "to appear", that no infinite expansion of $1$ in a Pisot base has arbitrarily long zero runs; the general exclusion of every Pisot number, $L(q)>0$, is attributed to this paper by [EJK98] (p. 202, result (c)). Remark 2 there: the gaps do tend to $0$ for $q=2^{1/m}$, $m\ge2$. [EJK98] then records the 1998 state: "We do not know whether $L(q)=0$ for all $q$ sufficiently close to 1" (p. 206), with Theorem 4 ($L(q)\le(q^2-1)e$ for $1<q<2$, so $L(q)\to0$ as $q\to1$) and Theorem 5 (if $1<q<\sqrt2$ and $l(q^2)=0$ then $L(q)=0$; in particular for every transcendental $q<\sqrt2$). The Pisot characterizations of the denser sequences are context: Bugeaud's (second-hand) and the Erdős--Joó--Schnitzer Theorem (p. 95), for $1<q<(1+\sqrt5)/2$, $q$ is Pisot iff $l_2(q)>0$ for the sums with digits $0,1,2$; neither decides the problem: for the problem's sums, a subset of theirs, they give gaps bounded away from $0$ only at Pisot $q$, all at least $q_0\approx1.3247$.

The first resolution (Erdős and Komornik's Theorem IV). Theorem IV of [ErKo98] (p. 59): "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 $m\ge1$", where $(y_k)=(y_k^{q,m})$ is the increasing sequence of the sums $\varepsilon_0+\varepsilon_1q+\cdots+\varepsilon_nq^n$ with $\varepsilon_i\in\{0,1,\ldots,m\}$ (pp. 58--59); for $m=1$ it is the problem's sequence. The introduction (p. 57) cites the question as "raised in [4], Problem 4", the 1990 Bulletin paper's Problem 4 quoted above, states "One of the purposes of this paper is to give an affirmative answer to this question", and gives the special case "$y_{k+1}-y_k\to0$ for all $q$ between 1 and $2^{1/4}$, except possibly the square root of the second Pisot number $\sqrt{p_2}\approx1.175$"; Remark (a) after the theorem says the property "probably" holds at $\sqrt{p_2}$ too and that the first author's death ended the study. The paper's $p_1,p_2$ are the site's $q_0,q_1$. The proof (pp. 77--78) applies the paper's Lemma 3.2, which turns a finite accumulation point of the difference set of one digit pattern and bounded gaps of two others into gaps tending to $0$ for their sum: for $q<2^{1/4}$ with $q^2$ not Pisot, the pattern of even powers is the sequence for $q^2$, whose difference set has a finite accumulation point by the paper's Theorem I (b) (because $q^2<(1+\sqrt5)/2$), and the patterns supported on $i\equiv1$ and $i\equiv3\pmod4$ have bounded gaps by Lemma 3.1, which needs $q^4\le2$, the source of the bound $2^{1/4}$; for $q=\sqrt{p_1}$ the same runs with period $3$ through $q^3\approx1.525$, which is not Pisot ("between the fifth and sixth Pisot numbers") and below $(1+\sqrt5)/2$. Only $\sqrt{p_2}$ is left out. Basis: the statement, the remarks and the proof are checked as far as the proof's reductions to Theorem I (b) and Lemmas 3.1 and 3.2; those results are not checked. The result page records one filing observation (the proof's first case is written for $q<2^{1/4}$ while the theorem allows equality, where the same argument applies). Acceptance: a refereed journal article (Acta Math. Hungar. 1998), cited by [Fe16] and the site as the first resolution.

The answer again (Feng's Theorem 1.4 with the authored deduction). Theorem 1.4 of [Fe16] (p. 3): "If $1<q<\sqrt{m+1}$ and $q^2$ is not a Pisot number, then $L_m(q)=0$. In particular, if $q\in(1,\sqrt2)$ and $q^2$ is not a Pisot number, then $L_1(q)=0$", where $L_1(q)=\limsup(x_{n+1}-x_n)$ for the problem's sequence; the paper derives it in one line from its Corollary 1.3 ($\ell_m(q)=0$ iff $q<m+1$ and $q$ is not Pisot, a corollary of the main density theorem) and the implication $\ell_m(q^2)=0\Rightarrow L_m(q)=0$ (Akiyama and Komornik's Lemma 2.5, first proved for $m=1$ as Theorem 5 of [EJK98]). Deduction made here: the smallest Pisot number is $q_0\approx1.3247$ ([Si44]; the site says the same), so no Pisot number lies in $(1,q_0)$; for $1<q<\sqrt{q_0}$ one has $q^2\in(1,q_0)$, hence $q^2$ is not Pisot, and $q<\sqrt{q_0}<\sqrt2$; Theorem 1.4 with $m=1$ gives $L_1(q)=0$, that is, $x_{k+1}-x_k\to0$. Numerically $1.1509^2\approx1.3246<q_0<1.3248\approx1.1510^2$, so $\sqrt{q_0}\approx1.1510$ and every $\epsilon\le\sqrt{q_0}-1\approx0.151$ answers the question. Theorem 1.4 decides every $q\in(1,\sqrt2)$ except the square roots of Pisot numbers ($\sqrt{q_0}\approx1.1510$, $\sqrt{q_1}\approx1.175$, $\ldots$), where it is silent; [ErKo98]'s Theorem IV covers $\sqrt{q_0}$ by its separate case (p. 78) and leaves $\sqrt{q_1}$ out below $2^{1/4}$, a point that Akiyama and Komornik's Theorem 1.4 (i) closes (next paragraph). Acceptance: a refereed journal article (JEMS 2016); the site's label; the thread's identification. Basis: claims checked for Theorem 1.4, Corollary 1.3, Theorem 1.2 and the derivation sentence; Feng's proof (Theorem 1.6 by way of Theorem 1.11, pp. 5--6, and Section 2, pp. 6--11) and the cited implication are not checked; the deduction above is the compilation's and is not independently reviewed.

The range extended (Akiyama and Komornik's Theorem 1.4 (i), from the arXiv text). Theorem 1.4 of Akiyama and Komornik, Discrete spectra and Pisot numbers (J. Number Theory 133 (2013), no. 2, 375--390; arXiv:1103.4508v1 of 23 March 2011, p. 4), states for a non-Pisot $1<q<2$: (i) if $1<q\le2^{1/3}\approx1.26$ then $L_1(q)=0$; (ii) if $1<q\le\sqrt2$ then $\ell_1(q)=L_2(q)=0$; (iii) $\ell_2(q)=L_3(q)=0$. Since every $q$ in $(1,2^{1/3}]$ is below $q_0\approx1.3247$, the non-Pisot hypothesis holds throughout part (i), so the gaps tend to $0$ for every $1<q\le2^{1/3}$, and the paper says that part (i) improves [ErKo98]'s Theorem IV, whose excluded point $\sqrt{q_1}$ lies in the new range. Feng's p. 3 reports the same: $L_1(q)=0$ for $1<q\le2^{1/3}$, cited to [ErKo98] and this paper together. The theorem is an accepted full claim on Akiyama and Komornik's page; the journal text is not compared with the arXiv text, and the paper is not held in the library.

Beyond the question (context, not the problem). Feng's Corollary 1.3 gives the lower limit exactly: $\liminf(x_{n+1}-x_n)=0$ for every non-Pisot $q\in(1,2)$ and for no Pisot $q$ (the site's first Feng sentence). For the upper limit, Feng's p. 3 reports Komornik's conjecture that $L_1(q)=0$ for every non-Pisot $q$ below the golden ratio and calls the general $L_m(q)$ question open; the characterization asked for in the first sentence of [EJK90]'s Problem 4 is therefore open for the lim sup. The thread's first comment links a document on a data repository that its poster describes as an unverified proof: a two-page note, "A Short Proof for Erdos Problem 1096", by Pedro Acosta De León, dated 16 April 2026, whose Theorem 1 is the deduction above (gaps tending to $0$ for every $q\in(1,\sqrt{q_0})$, from Feng's Theorem 1.4 and the minimality of $q_0$) and nothing further; it is a dated manuscript with a named author, so it has a claim page, pending, on Acosta De León's page; it was never filed on the proof-claims tab and carries no acceptance evidence.

Search scope. None of the routes below found a dispute of Feng's theorem or of the site's account, an open copy of [ErKo98], or a resolution of the lim sup question in general; [GWNT91] was located on the conference site.

• The site: problem page, discussion thread and proof-claims tab; the formal-conjectures file at the pinned commit and the external Lean file at its pinned commit (statement and closing lines); the community database, accessed 2026-09-18.
• The primary sources: [Fe16] pp. 1--3; [EJK90] pp. 377, 386, 387, 389 and 390; [EJS96] pp. 95 and 99; [EJK98] pp. 201--202 and 206--207; [ErKo98] pp. 57--60 and 74--78.
• arXiv: the abstract page of 1109.1407 (three versions, the last of 1 February 2015, "to appear in J. Eur. Math. Soc", no journal reference) and its API record; the search abs:Pisot AND (abs:spectrum OR abs:spectra OR abs:"non-integer base" OR abs:"non-integer bases") sorted by date (35 records, scanned by title; none on the gap question).
• Crossref: the records of [Fe16], [ErKo98], [EJK98] and, by bibliographic query, [EJK90] (DOI 10.24033/bsmf.2151) and [Bu96] (DOI 10.1007/BF00058941); no record for [EJS96].
• Semantic Scholar: the citation lists of [Fe16] (44 records) and [ErKo98] (64 records), scanned by title; the 2025--2026 items concern spectra of $m$-bonacci and other Pisot numbers, Delone sets and iterated function systems, none the problem's gap question.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: [Bu96], [GWNT91] (located), [Si44], Akiyama--Komornik's paper (quoted from its arXiv text), the two 1990 sources "to appear" behind [EJK90]'s Pisot recall.

Remaining gaps. (1) The first resolution, [ErKo98]: its Theorem IV is checked with its proof; the two second-hand ranges are reconciled (the printed range is Feng's, the site's its part below $\sqrt{q_1}$), and the field rests on that theorem, on Akiyama and Komornik's Theorem 1.4 (i), which closes the excluded point, and on Feng's theorem with Siegel's classical theorem, all first-hand. The proof's supports, the paper's Theorem I (b) and Lemma 3.2, are not checked. (2) The 1990 Bulletin paper's Problem 4 is the earliest located statement of the question; the 1991 set's 91:18 (p. 16) attributes the question and the $q_0$ guess to Erdős and Joó. (3) The Lean artifact's imported module, where the mathematics lives, is not examined, and nothing is built. (4) [Bu96] is not held; its theorem is quoted from the site and from [EJS96]. (5) Proof coverage: claims checked for Feng's Theorem 1.4 and the 1990, 1996 and 1998 statements; the one-page proof of [ErKo98]'s Theorem IV is checked as far as its reductions, and no other proof is checked; the deduction from Theorem 1.4 to the problem is the compilation's one line. (6) The thread's note rests on Feng's theorem and carries no acceptance evidence. (7) The general characterization of the $q$ with $x_{k+1}-x_k\to0$ (the first sentence of the 1990 Problem 4) is open for the lim sup, by Feng's 2016 account.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1990_characterization_unique_expansions_related_problems
• erdos_1990_characterization_unique_expansions_related_problems / problem_4
• erdos_1990_characterization_unique_expansions_related_problems / theorem_3
• erdos_1990_characterization_unique_expansions_related_problems / theorem_4
• erdos_1996_pisot_numbers
• erdos_1996_pisot_numbers / theorem
• erdos_1998_sequence_numbers_form_sums_powers_q
• erdos_1998_sequence_numbers_form_sums_powers_q / theorem_4
• erdos_1998_sequence_numbers_form_sums_powers_q / theorem_5
• erdos_komornik_1998_developments_non_integer_bases
• erdos_komornik_1998_developments_non_integer_bases / theorem_i
• erdos_komornik_1998_developments_non_integer_bases / theorem_ii
• erdos_komornik_1998_developments_non_integer_bases / theorem_iii
• erdos_komornik_1998_developments_non_integer_bases / theorem_iv
• erdos_komornik_1998_developments_non_integer_bases / theorem_v
• feng_2016_topology_polynomials_bounded_integer_coefficients
• feng_2016_topology_polynomials_bounded_integer_coefficients / corollary_1_3
• feng_2016_topology_polynomials_bounded_integer_coefficients / theorem_1_11
• feng_2016_topology_polynomials_bounded_integer_coefficients / theorem_1_2
• feng_2016_topology_polynomials_bounded_integer_coefficients / theorem_1_4
• feng_2016_topology_polynomials_bounded_integer_coefficients / theorem_1_6
• guy_1991_western_number_theory_problems
• guy_1991_western_number_theory_problems / problem_91_18