On the number of odd binomial coefficients
Let \(F(N)\) denote the sum-of-digit function \(\sum_{n<N} 2^{s(n)}\), where \(s(n)\) is the sum of the digits of the base 2-expansion of \(n\). The function \(F(N)\) counts the number of odd trinomial coefficients in the first \(N\) rows of Pascal's triangle. The behaviour of \({F(N)\over N^\theta}\), where \(\theta= {\log 3 \over \log 2}\), is described by a 1-periodic continuous function \(f\) of the dyadic logarithm of \(N\). The paper under review goes deeper into some properties of this classical function. The function \(f\) is proved to be differentiable almost everywhere (the values when \(f\) is differentiable are explicitly determined); \(f\) is shown to be nowhere monotonic. Furthermore, the values \(y\) for which the function \(f\) reaches its minimum are proved to be irrational: the binary representation of such a value \(g\) is a Sturmian sequence of angle \(\Theta\). One can thus deduce the transcendence of such a \(y\) from a result of \textit{S. Ferenczi} and \textit{C. Mauduit} [J. Number Theory (to appear)]. Finally, the author defines an algorithm which is conjectured to converge to the minimum of the function \(f\), provided a diophantine approximation condition involving \(\Theta\) is fulfilled.
scientific article; zbMATH DE number 897257