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Statement

Here $\{x\}$ is the fractional part and $\|x\|=\min(\{x\},1-\{x\})$ the distance to the nearest integer.

Theorem 1 (p. 136): "Let $\nu$ be a fixed real number, and let $r$ be a fixed positive number. If $t_0<t_1<t_2<\cdots$ is a sequence of positive real numbers satisfying $t_{n+1}\ge(1+r^{-1})t_n$ for $n=0,1,2,\ldots$ then there is a positive number $\xi$ such that $\{\xi t_n+\nu\}\le\min(r,1-2(3r+6)^{-2})$ for each integer $n\ge0$."

So the fractional parts $\{\xi t_n\}$ all avoid a subinterval of $[0,1)$ of length $c(r)=\max(1-r,2(3r+6)^{-2})$ (abstract). The paper draws on p. 137: "By Theorem 1, there is a positive number $\xi$ such that $\|\xi t_n\|\ge1/9(r+2)^2$", that is, with the shift $\nu$ centering the avoided interval on the integers, every $\xi t_n$ stays at distance at least $1/(9(r+2)^2)$ from $\mathbb Z$; consequently the graph on $\mathbb R$ with edges at the distances $t_n$ has chromatic number at most $9(r+2)^2$ for $r\ge1$. For a lacunary sequence of positive integers with ratio at least $1+\epsilon$ take $r=\epsilon^{-1}$: the separation is of order $\epsilon^2$ with no logarithmic factor. The theorem produces a positive real $\xi$ and does not assert that it is irrational.

Source. A. Dubickas, On the fractional parts of lacunary sequences, Math. Scand. 99 (2006), no. 1, 136--146, doi:10.7146/math.scand.a-15004; Theorem 1 on printed p. 136 (PDF p. 1 of the journal PDF), the consequences on p. 137 (PDF p. 2), read in the text layer and checked on the rendered page images. The artifact is identified in the source digest.

Read depth. Claims checked: the statement and the p. 137 consequences were read clause by clause on the page images. The proof (Sections 3 and 4) was read for its structure only and not checked.

Proof pointer

Two parts (p. 138). The bound $\{\xi t_n+\nu\}\le r$ follows from Theorem 4 (p. 138, proved in Section 3, p. 139): for any increasing sequence of positive reals there is $\xi>0$ with $\{\xi t_n+\nu\}\le t_n\sum_{j>n}t_j^{-1}$ for all $n$, by nested closed intervals $[(k_n-\nu)/t_n,(k_n-\nu+T_n)/t_n]$; under the ratio hypothesis the sum is at most $\sum_{j\ge1}(1+r^{-1})^{-j}=r$. The bound $1-2(3r+6)^{-2}$, needed for $r\ge r_0=0.9748\ldots$ (the root of $r=1-2(3r+6)^{-2}$), is proved in Section 4 (pp. 140--142) by a nested-interval construction in blocks of $g=[(7/2)(r+1)\log(r+2)]+1$ indices with $w=(2/9)(r+2)^{-2}$, removing at most $g+rt_{g(m+1)}|I_m|$ short intervals from each block interval and checking that a subinterval of the required length survives, which reduces to an inequality in $r$ verified numerically for $r\ge0.97$ (maximum of the left side about $0.9907$ near $r=12.2$). The method is that of Akhunzhanov and Moshchevitin, going back to de Mathan, Katznelson and Pollington (p. 138). Not reconstructed here.

Dependencies

Self-contained; the paper attributes the method to its references [3], [9], [16], [20] and cites Ruzsa, Tuza and Voigt [22] for the step from a separation $\|\xi t_n\|\ge q^{-1}$ to the chromatic bound $q$.

Bears on

• Problem 464: an explicit separation $\|\xi n_k\|\ge1/(9(r+2)^2)$, $r=\epsilon^{-1}$, for the corrected formulation, the "Dubickas" step in the site's list of improvements between Katznelson and Peres--Schlag; not the irrationality clause.
• Problem 894: restricted to the integers, the chromatic bound $9(r+2)^2$ colors the lacunary difference graph with $O(\epsilon^{-2})$ colors.