On pseudo-polynomials divisible only by a sparse set of primes and $\a$-primary pseudo-polynomials

TL;DR

This paper investigates the structure of pseudo-polynomials, constructing examples with prime divisibility restricted to sparse sets and analyzing the conditions under which $$-primary pseudo-polynomials are actual polynomials.

Contribution

It introduces constructions of pseudo-polynomials with prime divisibility confined to sparse sets and characterizes $$-primary pseudo-polynomials as polynomials under growth constraints.

Findings

Pseudo-polynomials can be constructed with prime divisibility only in sparse sets.

Not all pseudo-polynomials satisfy previous assumptions in the literature.

Under certain growth conditions, $$-primary pseudo-polynomials are polynomials.

Abstract

We explore two questions about pseudo-polynomials, which are functions $f:\mathbb N \to \mathbb Z$ such that $k$ divides $f(n+k) - f(n)$ for all $n,k$. First, for certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with $p|f(n)$ for some $n$ only if $p \in R$. This implies that not all pseudo-polynomials satisfy an assumption of a recent paper of Kowalski and Soundararajan. We also consider $\alpha$-primary pseudo-polynomials, where the pseudo-polynomial condition is only required for $k$ lying in a set of primes of density $\alpha$. We show that if an $\alpha$-primary pseudo-polynomial is $O(e^{(2/3-\epsilon) n})$, then it is a polynomial.