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Conference in memory of A. A. Karatsuba on number theory and applications
January 31, 2014 17:05–17:30, Moscow, Steklov Mathematical Institute, Lecture Room 530

On which value of the height the distribution of algebraic numbers is regular?

V. I. Bernik
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V. I. Bernik

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Abstract: In 1970, Baker and Schmidt introduced the definition of regular systems and proved the regularity of real algebraic numbers of any degree. This gives the possibility of obtaining the lower bound for the Hausdorff dimension of the set of real numbers which are approximated by algebraic numbers with a given order of approximation. The result of Baker and Schmidt was improved by Bernik in 1989. In 1999, Beresnevich proved the regularity of real algebraic numbers and his result was the best possible for functions defined on the set of algebraic numbers.
In his book “Approximation by algebraic numbers” (2004) Bugeaud notes that there is an important unsolved problem in the distribution of algebraic numbers to find the exact relation between the height of algebraic numbers and length of the intervals at which we can guarantee the existence of real algebraic numbers belonging to the interval. In the common work of the speaker, N.V. Budarina and H. O'Donnell this relation is found for the real algebraic numbers of third degree.
In the talk we will discuss the new results on the distribution of complex algebraic numbers in the circles of small radius obtained by the speaker in his common paper with N.V. Budarina and F.Gotze.

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