Termsuniform

Termsuniform is a coinage used in some mathematical discussions to describe a property of a sequence or collection of terms in which the terms are distributed uniformly across a defined set. In practice, the term is used to express that, as the sequence progresses, each possible term or residue occurs with approximately equal frequency, or equal probability, in the limit.

Formal notion: Let A be a finite set, and let (t_k) be a sequence with values in A. The sequence is termed termsuniform if, for every a in A, the limit as N→∞ of (1/N) times the number of k ≤ N with t_k = a equals 1/|A|. Similarly, for residues modulo m, the sequence of residues r_k = t_k mod m is termsuniform if each residue class occurs with limiting frequency 1/m. This concept is closely related to ideas of equidistribution and uniform distribution modulo m, and it is often discussed in the context of generators, samplers, or combinatorial constructions.

Notes on usage: Termsuniform is not a universally standardized term across all communities. When used, it typically serves as shorthand to denote uniformity properties of a sequence’s outputs, rather than as a formal, axiomatized concept.

Applications: The notion is relevant to the design and analysis of pseudorandom number generators, hashing schemes, randomized algorithms, and Monte Carlo methods, where unbiased or uniform sampling of a finite set or residue classes is desirable.

See also: Uniform distribution, Equidistribution, Pseudorandom number generator, Low-discrepancy sequence.