Title: Randomness Criterion $\Sigma$ and Its Applications

Author(s): Teturo Kamae, Dong Han Kim and Yu-Mei Xue
Issue: Volume 80 Series A Part 2 Year 2018
Pages: 356 -- 384
The Sigma function, which is the sum of the squares of the number of occurrences of every factor, is a criterion of randomness, measuring specially the uniformity of the block distribution. An infinite word whose prefixes attain asymptotically the smallest possible value of it is called Sigma-random. We prove that the Champernowne word is Sigma-random. We also consider less complex words which have values with asymptotically larger order, Sturmian words and almost $0$-words.
AMS (2000) subject classification. Primary 65C10; Secondary 11K45, 68R15.
Keywords and phrases: Randomness criterion, Champernowne number, Sturmian word