The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem arises across a wide range of topics in computer science, including loop termination, (weighted) automata theory, and the analysis of linear dynamical systems, amongst many others. Decidability of the Skolem Problem is notoriously open. The state of the art is a decision procedure for recurrences of order at most 4: an advance achieved some 40 years ago based on Baker's theorem on linear forms in logarithms of algebraic numbers. Recently, a new approach to the Skolem Problem was initiated via the notion of a Universal Skolem Set: a set $\mathbf{S}$ of positive integers such that it is decidable whether a given non-degenerate linear recurrence sequence has a zero in $\mathbf{S}$. Clearly, proving decidability of the Skolem Problem is equivalent to showing that $\mathbb{N}$ is a Universal Skolem Set. The main contribution of the present paper is to exhibit a Universal Skolem Set of positive density that moreover has density one subject to the Bateman-Horn conjecture in number theory. The latter is a central unifying hypothesis concerning the frequency of prime numbers among the values of systems of polynomials, and provides a far-reaching generalisation of many classical results and conjectures on the distribution of primes.

翻译：Skolem问题旨在判断给定整数线性递归序列是否存在零项。该问题广泛出现在计算机科学各领域中，包括循环终止性、(加权)自动机理论、线性动力系统分析等诸多方面。Skolem问题的可判定性至今仍是公开难题。目前最前沿的成果是适用于至多4阶递归序列的判定程序：这一突破性进展约在40年前基于Baker关于代数数对数线性形式的定理而实现。近年来，通过通用Skolem集概念为Skolem问题开辟了新途径：若存在正整数集合 **S**，使得给定非退化线性递归序列在 **S** 中是否存在零项是可判定的，则称 **S** 为通用Skolem集。显然，证明Skolem问题的可判定性等价于证明自然数集 **N** 是通用Skolem集。本文主要贡献在于构造了一个具有正密度的通用Skolem集，该集合在数论中的Bateman-Horn猜想成立条件下密度为1。Bateman-Horn猜想是关于多项式系统取值为素数频率的核心统一假说，它深远地推广了素数分布领域的诸多经典结果与猜想。