Dubickas 2006 fractional parts lacunary sequences

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theorem_1: Dubickas's 2006 theorem that for every real shift nu and every lacunary sequence of positive reals with consecutive ratios at least 1 + 1/r there is a positive multiplier xi whose shifted fractional parts all lie below min(r, 1 - 2(3r+6)^(-2)); with the shift chosen suitably it gives ||xi t_n|| at least 1/(9(r+2)^2) for all n, a separation of order epsilon squared for ratio 1 + epsilon.

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Dubickas, Artūras, On the fractional parts of lacunary sequences. Math. Scand. 99 (2006), no. 1, 136--146; DOI 10.7146/math.scand.a-15004 (Crossref record, issued 1 September 2006). Received 20 July 2005. The site's key Du06.

The copy read for this card is the journal's typeset article (Acrobat Distiller, 2006), eleven pages with the printed folios 136--146 (PDF p. $n$ is printed p. $135+n$) and a text layer that drops superscripts, so exponents were checked on the rendered page images: the Akhunzhanov--Moshchevitin bound on p. 137 reads $2^7r^2$ on the page and "27 r^2" in the text layer. Page references are printed pages.

Read status: claims checked for Theorem 1, the p. 137 chromatic-number paragraph, Corollary 2 and Corollary 3 (read clause by clause; pp. 136--137 on the page images); Theorem 4 and the digit theorems (Theorems 5, 8, 9 and Corollaries 6--7) were read as statements in the text layer; the proofs (Sections 3--6) were not checked.

Dubickas proves Theorem 1 (p. 136): for any real $\nu$ and any lacunary sequence $t_0<t_1<\cdots$ of positive reals with $t_{n+1}\ge(1+r^{-1})t_n$, $r>0$ fixed, there exists $\xi>0$ with $\{\xi t_n+\nu\}\le\min(r,1-2(3r+6)^{-2})$ for all $n\ge0$, so all the fractional parts $\{\xi t_n\}$ miss a subinterval of $[0,1)$ of length $c(r)=\max(1-r,2(3r+6)^{-2})$. The first inequality comes from Theorem 4 (p. 138), a nested-interval statement for arbitrary increasing sequences; the second, for $r\ge0.97$, is proved in Section 4 (pp. 140--142) with nested intervals, the method of Akhunzhanov and Moshchevitin [3], which the paper traces back to de Mathan [9], Katznelson [16] and Pollington [20] (p. 138). Corollaries 2 and 3 (p. 137) specialize to $t_n=n^\lambda a^n$ and to the distance-to-nearest-integer form $|\xi a^n|\ge\max((1-r)/2, (3r+6)^{-2})$ with $r=1/(a-1)$; Corollary 2 contains Tijdeman's bound $\{\xi a^n\}\le1/(a-1)$. The introduction (p. 137) says that "Erdős [14] raised an interesting question in this direction which was answered independently by Pollington [20] and de Mathan [9]", [14] being Erdős, Problems and results on Diophantine approximations. II, Répartition modulo 1 (Marseille-Luminy 1974), Lecture Notes in Math. 475 (1975), 89--99, and that Erdős noticed the connection with the chromatic number of the Cayley graph, studied by Katznelson [16] and by Ruzsa, Tuza and Voigt [22]. The chromatic application (p. 137): for a lacunary $T=\{t_0=1,t_1,\ldots\}$ with ratio at least $1+r^{-1}$ let $G$ be the graph whose vertices are the real numbers, $x$ and $y$ adjacent when $|x-y|\in T$; [22] proved $\chi(G)\ge r$ and an upper bound, Akhunzhanov and Moshchevitin [3] improved the upper bound to $\chi(G)\le2^7r^2$ for $r\ge3$, and since $\chi(G)\le q$ follows from a $\xi$ with $\|\xi t_n\|\ge q^{-1}$ for all $n$ (as shown in [22]), Theorem 1 gives $\|\xi t_n\|\ge1/(9(r+2)^2)$ and hence $\chi(G)\le9(r+2)^2$ for every $r\ge1$. A further application (Theorem 5, p. 139) expresses $\sqrt{10}$ as a ratio $y/x$ of positive reals whose digits after the decimal point all lie in $\{0,1,2,3,4\}$; Section 6 treats $t_n=n!$ and other fast-growing sequences.

Source: https://doi.org/10.7146/math.scand.a-15004. No notice is printed in the article beyond the header "MATH. SCAND. 99 (2006), 136-146"; the journal's article page (https://journals.msp.org/mscand/article/view/693, read 2026-10-02) states no license, and the publisher's policy page (https://msp.org/publications/policies/, read 2026-10-02) says the publisher "must at least obtain an exclusive license for all commercial distribution of the published version of record", allows CC-BY only "if strictly required by the funder", and makes articles "free to access and read after six years past publication", every other right reserved.

Compiled scope

Pages 136--137 were read on the page images and the rest in the text layer; Theorem 1 is compiled as a statement with a proof pointer; no proof step was checked and nothing here is independently reviewed. The graph $G$ of p. 137 is on the real line with a real lacunary distance set; the site's Problem 894 concerns the induced subgraph on the integers with an integer sequence, and Problem 464 the Diophantine question rather than the chromatic one.

Bears on. #464, for whose corrected formulation (fractional parts of $\theta n_k$ not dense modulo $1$) Theorem 1 gives the explicit separation $\|\xi t_n\|\ge1/(9(r+2)^2)$, of order $\epsilon^2$ for ratio $1+\epsilon$ ($r=\epsilon^{-1}$), without a logarithm; the paper produces a positive real $\xi$ and does not assert its irrationality, so it does not by itself settle the irrational question; p. 137 attests the original solutions of Pollington and de Mathan and identifies the 1975 chapter in which Erdős posed the question. #894, for which the chromatic bound $\chi(G)\le9(r+2)^2$ on the real line restricts to the integers and gives a coloring with $O(\epsilon^{-2})$ colors, the "see also Dubickas" step in Peres and Schlag's history of the bound.

Results.

• Theorem 1 (p. 136): For fixed real $\nu$ and $r>0$, any sequence of positive reals with $t_{n+1}\ge(1+r^{-1})t_n$ admits $\xi>0$ with $\{\xi t_n+\nu\}\le\min(r,1-2(3r+6)^{-2})$ for all $n\ge0$.
• Chromatic bound (p. 137): For the graph on $\mathbb R$ with edges at the lacunary distances $t_n$, $t_{n+1}/t_n\ge1+r^{-1}$, $\chi(G)\le9(r+2)^2$ for $r\ge1$, improving $\chi(G)\le2^7r^2$ ($r\ge3$).
• Corollary 2 (p. 137): For $a>1$, $\lambda\ge0$, $r=1/(a-1)$ and every subinterval $I=(s,s+\max(1-r,2(3r+6)^{-2}))$ of $[0,1)$ there is $\xi>0$ with all $\{\xi n^\lambda a^n\}$, $n\ge0$, outside $I$.
• Corollary 3 (p. 137): For $a>1$ and $r=1/(a-1)$ there is $\xi>0$ with $\|\xi a^n\|\ge\max((1-r)/2,(3r+6)^{-2})$ for all $n$.
• Digit application (Theorem 5, p. 139): $\sqrt{10}=y/x$ for some positive reals $x$ and $y$ whose digits after the decimal point all lie in $\{0,1,2,3,4\}$.

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