Uniformnos

Uniformnos is a term used in some mathematical discussions to denote real numbers that exhibit uniformity in the distribution of their digit blocks across representations in a base. The concept is closely related to the classical idea of normal numbers, and in certain contexts serves as a synonym or specialization of that notion.

Definition: A real number x in the interval [0,1) is a base-b uniformno if for every finite string s of digits from 0 to b−1, the frequency of s among the first N digits of x’s base-b expansion tends to 1/b^|s| as N grows. If this condition holds for every base b ≥ 2, x is called absolutely uniformno (often referred to as an absolutely normal number).

Properties and examples: Almost every real number is a uniformno in the measure-theoretic sense, by the standard results on normal numbers. Constructing explicit uniformnos can be challenging; many classical examples, like the Champernowne constant, are known to be normal in base 10 (hence a uniformno in that base) but may not be normal in other bases. The existence and construction of numbers that are uniformnos in all bases (absolute uniformnos) are subjects of ongoing study in number theory.

Applications and context: The study of uniformnos intersects with randomness tests, pseudo-random number generation, and questions about digit distribution in computational mathematics. It provides a framework for discussing how uniformly information is spread across digits and blocks in numeral systems.

See also: Normal number, Absolutely normal number, Equidistribution, Champernowne constant.