THE SET OF INCOMPLETE SUMS OF THE MODIFIED GUTHRIE-NYMANN SERIES

Abstract

In this paper we study topological and metric properties of the set of incomplete sums for positive series $\sum {a_k}$, where $a_{2n-1}=3/4^n+3/4^{in}$ and $a_{2n}=2/4^n+2/4^{in}$, $n \in N$. The series depends on positive integer parameter $i \geq 2$ and it is some perturbation of the known Guthrie-Nymann series. We prove that the set of incomplete sums of this series is a Cantorval (which is a specific union of a perfect nowhere dense set of zero Lebesgue measure and an infinite union of intervals), and its Lebesgue measure is given by formula: $\lambda(X^+_i)=1+\frac{1}{4^i-3}.$ The main idea of ??proving the theorem is based on the well-known Kakey theorem, the closedness of sets of incomplete sums of the series and the density of the set everywhere in a certain segment. The work provides a full justification of the facts for the case $i=2$. To justify the main facts, the ratio between the members and the remainders of the series is used. For $i=2$ we have $r_0=\sum {a_k}=2$, $a_{2n}-r_{2n}= \frac{1}{3} \cdot \frac{1}{4^n} + \frac{5}{3} \cdot \frac{1}{16^n}$ $r_{2n-1}-a_{2n-1}= \frac{2}{3} \cdot \frac{ 1}{4^n}-\frac{2}{3} \cdot \frac{1}{16^n}$. The relevance of the study of the object is dictated by the problems of the geometry of numerical series, fractal analysis and fractal geometry of one-dimensional objects and the theory of infinite Bernoulli convolutions, one of the problems of which is the problem of the singularity of the convolution of two singular distributions.