Fourier analysis of continued fractions with restricted quotients
The main nice result of this paper reads as follows. Let \(A\) be a finite set of positive integers containing at least two elements, and let \(F(A)\) be the set of irrational numbers in \((0,1)\) whose continued fraction expansion has all its partial quotients in \(A\). Then, if the Hausdorff dimension of \(F(A)\) is \(>{1\over 2}\), the set \(F(A)\) contains infinitely many normal numbers. If \(A:= \{1,2,\dots, N\}\) this improves on a result of \textit{R. Kaufman} (more precisely on a remark of Baker to a result of Kaufman) where the result holds true for \(N\geq 3\) (see [Mathematika 27, 262--267 (1980; Zbl 0455.10035)], see also [\textit{H. L. Montgomery}, Ten lectures on the interface between analytic number theory and harmonic analysis. Providence, RI: AMS (1994; Zbl 0814.11001)]). Please note that reference [3] has appeared [\textit{Y. Bugeaud}, C. R. Math., Acad. Sci. Paris 335, No. 2, 117--120 (2002; Zbl 1006.11043)].
continued fraction with bounded partial quotients
scientific article; zbMATH DE number 2109720