Seminario: "On the number of residues of certain second-order linear recurrences" - Carlo Sanna

Italy - France Analytic Number Theory Workshop

Università degli Studi di Genova, July 8-10 2024

DIMA - Dipartimento di Matematica

https://www.dima.unige.it/ant/italy-france/index.html

Abstract. For every monic polynomial f ∈ Z[X] with deg(f) ≥ 1, let L(f) be the set of all linear recurrences with values in Z and characteristic polynomial f, and let R(f) := { ρ(x; m) : x ∈ L(f), m ∈ Z^+}, where ρ(x; m) is the number of distinct residues of x modulo m. Dubickas and Novikas, motivated by some problems on fractional parts of powers of Pisot numbers, proved that R(X^2 − X − 1) = Z^+. We generalize this result by showing that R(X^2 − a_1 X − 1) = Z^+ for every nonzero integer a_1. As a corollary, we deduce that for all integers a_1 ≥ 1 and k ≥ 4 there exists ξ ∈ R such that the sequence of fractional parts {ξαn} , where α := (a_1 + sqrt(a_1^2 + 4)) /2, has exactly k limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.
This is a joint work with Federico Accossato.