Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits

• Дата: 2005
• Автори: Pratsiovytyi, M.V., Torbin, H.M.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2005
• Назва видання: Український математичний журнал
• Теми: Статті
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/165826
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits / M.V. Pratsiovytyi, H.M. Torbin // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1163–1170. — Бібліогр.: 8 назв. — англ.

Dedicated to V. S. Korolyuk on occasion of his 80-th birthday Properties of the set Tₛ of "particularly nonnormal numbers" of the unit interval are studied in details (Tₛ consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proven that the set Tₛ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ( Tₛ is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented.