On the b-ary expansions of log (1 + (1)/(a)) and \mathrm e

2015/10/01 by Yann Bugeaud, Bugeaud, Yann, Dong Han Kim +1
Mathematics · #11A63 #11J82 #68R15 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A63 #msc:11J82 #msc:68R15

paper · pdf · doi:10.48550/arxiv.1510.00282

17 pages

arxiv created 2015/10/01 · arxiv updated 2015/10/02

Abstract

Let b ≥ 2 be an integer and ξ an irrational real number. We prove that, if the irrationality exponent of ξ is equal to 2 or slightly greater than 2, then the b-ary expansion of ξ cannot be `too simple', in a suitable sense. Our result applies, among other classical numbers, to badly approximable numbers, non-zero rational powers of \mathrm e, and log (1 + (1)/(a)), provided that the integer a is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its b-ary expansion.

Related

• On the b-ary expansion of a real number whose irrationality exponent is close to 2
• On the irrationality exponent of real numbers with low complexity expansion
• How to approximate irrational numbers nicely?
• On the expansions of real numbers in two integer bases
• Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik