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Measures of sets decomposing the simply normal numbers in the unit interval
Author(s) -
Slivka John,
Severo Norman C.
Publication year - 1994
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300007269
Subject(s) - mathematics , countable set , interval (graph theory) , integer (computer science) , unit interval , real number , combinatorics , base (topology) , lebesgue integration , set (abstract data type) , discrete mathematics , borel set , lebesgue measure , unit (ring theory) , mathematical analysis , mathematics education , computer science , programming language
For any fixed positive real number ε, any integer b ≥2 and any d ε{0, 1,…, b−1}, the set of Borel's simply normal numbers to base b in [0, 1] is partitioned into a countable number of sets in eight different ways according to the largest place and the number of places at which the proportion d 's to that place in the b ‐adic expansion of such a number exceeds or is not less than b −1 – ε, or is less than or does not exceed b −1 – ε. For selected values ε, the Lebesgue measures of the sets in these decompositions are given explicitly.