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On the binary expansions of algebraic numbers
Abstract
Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number \#(vbar y vbar, N) of 1bits in the expansion of vbar y vbar through bit position N satisfies \#(vbar y vbar, N) > CN^1/D for a positive number C (depending on y) and sufficiently large N. This in itself establishes the transcendency of a class of reals \sum_n \geq 0 1/2^f(n) where theintegervalued function f grows sufficiently fast; say, faster than any fixed power of n. By these methods we reestablish the transcendency of the KempnerMahler number \sum_n \geq 01/2^2^n, yet we can also handle numbers with a substantially denser occurrence of 1's. Though the number z = \sum_n \geq 01/2^n^2 has too high a 1's density for application of our central result, we are able to invoke some rather intricate numbertheoretical analysis and extended computations to reveal aspects of the binary structure of z.
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