$G$-Sets and Linear Recurrences Modulo Primes

Numeration Systems, Linear Recurrences, and Regular Sets

A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2, . . . expresses a non-negative integer n as a sum n = ∑i j=0 ajuj . In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1, . . .) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the ...

Construction of Equidistributed Generators based on linear recurrences modulo 2

Construction of Equidistributed Generators Based on Linear Recurrences Modulo 2

Random number generators based on linear recurrences modulo 2 are widely used and appear in different forms, such as the simple and combined Tausworthe generators, the GFSR, and the twisted GFSR generators. Low-discrepancy point sets for quasi-Monte Carlo integration can also be constructed based on these linear recurrences. The quality of these generators or point sets is usually measured by c...

Periods of Orbits modulo Primes

Let S be a monoid of endomorphisms of a quasiprojective variety V defined over a global field K. We prove a lower bound for the size of the reduction modulo places of K of the orbit of any point α ∈ V (K) under the action of the endomorphisms from S. We also prove a similar result in the context of Drinfeld modules. Our results may be considered as dynamical variants of Artin’s primitive root c...

Some Curious Congruences modulo Primes

Let n be a positive odd integer and let p > n + 1 be a prime. We mainly derive the following congruence: ∑ 0<i1<···<in<p ( i1 3 ) (−1)i1 i1 · · · in ≡ 0 (mod p).