On the Derivative of the Minkowski Question-Mark Function

Author:

Gayfulin Dmitry1

Affiliation:

1. National Research University Higher School of Economics , Moscow , Russia

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.

Publisher

Walter de Gruyter GmbH

Subject

General Earth and Planetary Sciences,General Environmental Science

Reference22 articles.

1. [1] ALKAUSKAS, G.: Integral Transforms of the Minkowski Question Mark Function,PhD. Thesis, Nottingham, 2008.

2. [2] DENJOY, A.: Sur une fonction reelle de Minkowski C. R. Acad.Sci.Paris 194 (1932), 44–46.

3. [3] DENJOY, A.: Sur une fonction de Minkowski J. Math. Pures Appl. 17 (1938), 105–151.

4. [4] DUSHISTOVA, A. A.-MOSHCHEVITIN, N. G.: On the derivative of the Minkowski question mark function ?(x), Fundam. Prikl. Mat. 16 (2010), 33-44

5. translation in J. Math.Sci.(N.Y.) 182 (2012), 463-471. (In Russian)

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