Derivative of the Minkowski function for numbers with bounded partial quotients

Abstract

It is well known that the derivative of the Minkowski function ? ( x ) ?(x) (whenever exists) may take only two values: 0 0 and + ∞ +\infty . Let E n \textbf {E}_n be the set of irrational numbers on the interval [ 0 ; 1 ] [0; 1] whose partial quotients (related to the continued fraction expansion) do not exceed n n . It is also known that the quantity ? ′ ( x ) ?’(x) at a point x = [ 0 ; a 1 , a 2 , … , a t , … ] x=[0;a_1,a_2,\dots ,a_t,\dots ] is linked with the limit behavior of the arithmetic means ( a 1 + a 2 + ⋯ + a t ) / t (a_1+a_2+\dots +a_t)/t . In particular, A. Dushistova, I. Kan, and N. Moshchevitin showed that if x ∈ E n x\in \textbf {E}_n satisfies a 1 + a 2 + ⋯ + a t > ( κ 1 ( n ) − ε ) t a_1+a_2+\dots +a_t>(\kappa ^{(n)}_1-\varepsilon ) t , where ε > 0 \varepsilon >0 and κ 1 ( n ) \kappa ^{(n)}_1 is a certain explicit constant, then ? ′ ( x ) = + ∞ ?’(x)=+\infty . They also showed that the quantity κ 1 ( n ) \kappa ^{(n)}_1 cannot be increased. In the present paper, a dual problem is treated: how small may the quantity a 1 + a 2 + ⋯ + a t − κ 1 ( n ) t a_1+a_2+\dots +a_t-\kappa ^{(n)}_1 t be if ? ′ ( x ) = 0 ?’(x)=0 ? Optimal estimates in this problem are found.