On the Derivative of the Minkowski Question-Mark Function

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; a1,a2,...,at,...] is connected with the limit behaviour of the arithmetic mean (a1 +a2 +···+at)/t. Particularly, N. Moshchevitin and A. Dushistova showed that if a1+a2+⋯+at<κ1, {a_1} + {a_2} + \cdots + {a_t} < {\kappa _1}, where κ1=2log(1+52)/log2=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 κ1 is non-improvable. We consider a dual problem: how small can be the quantity a1 + a2 + ··· + at − κ1t if we know that ? (x) = 0? We obtain the non-improvable estimates of this quantity.