The joint distribution of the binary digits of integer multiples. (English) Zbl 0489.10008

Schmid, Johannes

Acta Arith. 43, 391-415 (1984).

Let \(B(n)\) denote the number of digits 1 in the binary representation of \(n\in \mathbb N_0\). Then for different odd \(k_1, k_2,\ldots, k_s\in \mathbb N\): \[ \# \{0\le n<x; \forall \nu=1, 2, \ldots, s: B(k_\nu n) = a_\nu\}= \] \[ = \frac{x}{\sqrt{2\pi \log x}^s \sqrt{\det V}} \exp\left(-\frac1{2\log x}\left(a - \frac{\log x}{2}\right) V^{-1}\left(a - \frac{\log x}{2}\right)'\right) + O\left(\frac{x}{\sqrt{\log x}^s\sqrt{\log x}}\right) \] with a positive-definite matrix \(V\). Herein, the \(O\)-constant does not depend on \(a:= (a_\nu)_{\nu =1}^s\in \mathbb Z^s\). \((\log := \) logarithm to base 2.) Analogous results hold for \[ \# \{0\le n<x; B(k_1n) - B(k_2n) =a\} \quad\text{and}\quad \# \{0\le n<x; \forall \nu\text{:} \frac{B(k_\nu n) - \log x/2}{\sqrt{\log x}} \le \xi_\nu\}. \]
[An earlier version of this article was written as a doctoral thesis under E. Wirsing at the University of Ulm.]

Reviewer: Johannes Schmid (Ulm)

MSC:

11A63	Radix representation; digital problems
11K16	Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11B05	Density, gaps, topology

Keywords:

binary digital sum; density of special sequences