Intervals of small measure containing an algebraic number of given height
Rational numbers are uniformly distributed, even though distances between rational neighbors in a Farey sequence can be quite different. This property doesn’t hold for algebraic numbers. In 2013 D. Koleda [6, 7] found the distribution function for real algebraic numbers of an arbitrary degree under their natural ordering. It can be proved that the quantity of real algebraic numbers of degree and height 𝐻(𝛼) 6 asymptotically equals 𝑐1(𝑛)𝑄𝑛+1. Recently it was proved that there exist intervals of length 𝑄-𝛾, > 1, free of algebraic numbers 𝛼,𝐻(𝛼) 6 𝑄, however for 0 6 < 1 there exist at least 𝑐2(𝑛)𝑄𝑛+1-𝛾 algebraic numbers in such intervals. In this paper we show that special intervals of length 𝑄-𝛾 may contain algebraic numbers even for large values of 𝛾, however their quantity doesn’t exceed 𝑐3𝑄𝑛+1-𝛾. An earlier result by A. Gusakova [16] was proved only for the case = 3 2 .