Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros
Alina Ostafe, Carl Pomerance, Igor E. Shparlinski
TL;DR
This paper studies the solvability of linear equations in elements of a fixed finitely generated subgroup $\Gamma$ of a number field and shows that such equations are solvable only for a small proportion of coefficient choices, using a modular method influenced by Erdős. It then transfers these ideas to bound the number of linear recurrence sequences with zeros, proving $Z_k(\Gamma,H) \le H^{d k - 1 + o(1)}$ and, for separable characteristic polynomials $f$, that $U^{k-1} \le Z_f(U) \le U^{k-1 + o(1)}$, with extensions to inhomogeneous cases and prescribed values. A key technical ingredient is constructing integers with unusually small Carmichael values $\lambda(n)$ from primes of positive relative density, via a set of primes with $p-1$ dividing a common multiple $M = \mathrm{lcm}(1,\dots,\lfloor y\rfloor)$ and applying a density/ Brun sieve argument combined with Chebotarev-type results. The results provide power-saving upper bounds and essentially tight estimates in the recurrence-sequence setting, highlighting the rarity of solvable $\mathcal{S}$-unit equations in large finite windows and zeros in linear recurrences.
Abstract
We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a classical idea of P. Erdős (1935). We then use similar ideas to get a tight upper bound on the number of linear recurrence sequences which attain a zero value.