On the binary expansions of algebraic numbers
Using concepts from additive number theory and results on binary evaluations and partial series, the authors prove that for a real algebraic number \(y\) of degree \(D> 1\) the number \(\#(|y|, N)\) of 1-bits in the expansion of \(|y|\) through bit position \(N\) is greater than \(CN^{1/D}\), where \(C> 0\) is a number depending only on \(y\), and \(N\) is sufficiently large. By means of this result the transcendency of a class of reals \(\sum_{n\geq 0} 2^{-f(n)}\) is obtained, where the integer-valued function \(f\) grows sufficiently fast. For example, for any real \(\alpha> 1\) the number \(\sum_{n\geq 0} 2^{-\lfloor\alpha^n\rfloor}\) is transcendental, and so the transcendency of the Kempner-Mahler number \(\sum_{n\geq 0} 2^{-2^n}\) is re-established. Further, they also study certain ``borderline'' numbers, and prove that the number \(\sum_{n\geq 0}2^{-n^2}\) is not a quadratic irrational. Moreover, some open problems are proposed.
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