ON ARITHMETIC PROGRESSIONS IN THE DIGITS OF ENGEL EXPANSIONS

Zhigang Tian () and Lulu Fang
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Zhigang Tian: School of Mathematics, South China University of Technology, Guangzhou 510640, P. R. China
Lulu Fang: School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

FRACTALS (fractals), 2025, vol. 33, issue 03, 1-9

Abstract: Denote by Eℱ and E℠the sets of real numbers whose Engel expansion digits contain arbitrarily long arithmetic progressions and arbitrarily long arithmetic progressions with arbitrary common difference, respectively. In this paper, the sizes of Eℱ and E℠are investigated from the measure-theoretical, fractal and topological viewpoints. We prove that Eℱ and E℠are of Lebesgue measure zero but have full Hausdorff dimension. Moreover, E℠is of first category while Eℱ is residual.

Keywords: Arithmetic Progressions; Engel Expansions; Lebesgue Measure; Baire Category; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X24501457

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