On the Derivative of the Minkowski Question-Mark Function

Abstract

Abstract The Minkowski question-mark function ?(x) is a continuous monotonous function defined on [0, 1] interval. It is well known fact that the derivative of this function, if exists, can take only two values: 0 and +∞.It isalso known that the value of the derivative ? (x)atthe point x =[0; a 1,a 2,...,a t ,...] is connected with the limit behaviour of the arithmetic mean (a 1 +a 2 +···+a t )/t. Particularly, N. Moshchevitin and A. Dushistova showed that if a 1 + a 2 + ⋯ + a t < κ 1 , {a_1} + {a_2} + \cdots + {a_t} < {\kappa _1}, where κ 1 = 2 log ( 1 + 5 2 ) / log 2 = 1.3884 … {\kappa _1} = 2\log \left( {{{1 + \sqrt 5 } \over 2}} \right)/\log 2 = 1.3884 \ldots , then ?′(x)=+∞.They also proved that the constant κ 1 is non-improvable. We consider a dual problem: how small can be the quantity a 1 + a 2 + ··· + a t − κ 1 t if we know that ? (x) = 0? We obtain the non-improvable estimates of this quantity.