On the Bits Counting Function of Real Numbers

Let Bn(x) denote the number of 1’s occuring in the binary expansion of an irrational number x > 0. A difficult problem is to provide non-trivial lower bounds for Bn(x) for interesting number numbers like √ 2, e or π: their conjectural simple normality in base 2 is equivalent to Bn(x) ∼ n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x, y > 0, which we prove to be sharp up to a multiplicative constant. As a byproduct, we provide an answer to a question raised by Bailey et al. (On the binary expansions of algebraic numbers, J. Théor. Nombres Bordeaux 16 (2004), no. 3, 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 non-trivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number.