Sequences that omit a box (modulo 1)
Let \(a_1<a_2<\dots\) be an increasing sequence of positive integers. For an open interval \(I\) in \([0,1)\) of length \(|I|\) denote the set \(X=\{x\in[0,1); a_nx\bmod 1\not\in I,\, n=1,2,\dots\}\). \textit{H. Weyl} [Math. Ann. 77, 313--352 (1916; JFM 46.0278.06)] proved that \(X\) has zero Lebesgue measure. \textit{M. D. Boshernitzan} [Adv. Math. 108, No. 1, 104--117 (1994; Zbl 0815.11013)] showed that the Hausdorff dimension \(\dim X=0\), whenever \(\frac{a_{n+1}}{a_n}\to 1\). Assuming \(a_n=O(n)\), both \textit{J.-P. Kahane} [Ann. Inst. Fourier 14, No. 2, 519--526 (1964; Zbl 0151.04402)] and \textit{Y. Amice} [Ann. Inst. Fourier 14, No. 2, 527--531 (1964; Zbl 0178.38003)] found that \(X\) is finite. In this paper the author extends the Kahane-Amice and Boshernitzan results to higher dimension. Take \(\mathbb I=I_1\times\dots\times I_d\) to be an open box in \([0,1)^d\) and denote by \(\mathbb X\) the set which corresponds to the above \(X\), then he proves \(\dim\mathbb X=d-1\). Define that an increasing sequence \(a_1<a_2<\dots\) of positive integers satisfies condition \(D(c)\) if there exists a sequence of intervals \([u_j,v_j]\), \(j=1,2,\dots\), such that \(v_j-u_j\to\infty\), \(1+v_j-u_j\leq c\#\{n=1,2,\dots;a_n\in[u_j,v_j]\}\). For integer sequences \(a^{(i)}_1<a^{(i)}_2<\dots\), \(i=1,2,\dots,d\), define \(\mathbb X\) to be the set of \(\mathbf x\in[0,1)^d\) for which \(a^{(1)}_{k_1}x_1\bmod 1\not\in I_1,\) \(\dots\), \(a^{(d)}_{k_d}x_d\bmod 1\not\in I_d\) for all \(a^{(1)}_{k_1}<\dots<a^{(d)}_{k_d}\). \textit{R. Kaufman} [Proc. Am. Math. Soc. 19, 1501--1503 (1968; Zbl 0179.06902)] proved that whenever \(a^{(i)}_1<a^{(i)}_2<\dots\), \(i=1,2,\dots,d\) satisfy \(D(c_i)\), then \(\mathbb X\) is countable. The author proves that \(\mathbb X\) is finite and \[ \#\mathbb X\leq\frac{18d(5d)^{2d}}{(|I_1|\dots|I_d|)^2}\max_{i}\frac{c_i}{|I_i|}. \]
scientific article; zbMATH DE number 5918247
scientific article