ON THE BITS COUNTING FUNCTION OF REAL NUMBERS
Tanguy Rivoal · Journal of the Australian Mathematical Society · 2008
Abstract LetBn(x) denote the number of 1’s occurring in the binary expansion of an irrational numberx>0. A difficult problem is to provide nontrivial lower bounds forBn(x) for interesting numbers such as $\sqrt {2}$ ,eorπ: their conjectural simple normality in base 2 is equivalent toBn(x)∼n/2. In this article, amongst other things, we prove inequalities relatingBn(x+y),Bn(xy) andBn(1/x) toBn(x) andBn(y) for any irrational numbersx,y>0, which we prove to be sharp up to a multiplicative constant. As a by-product, we provide an answer to a question raised by Baileyet al. (D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, ‘On the binary expansions of algebraic numbers’,J. Théor. Nombres Bordeaux16(3) (2004), 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides a nontrivial lower bound forBn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds forBn(x) whenxis a transcendental number.