Two classes of modular $p$-Stanley sequences

TL;DR

This paper introduces two new constructions of modular p-Stanley sequences for primes greater than 3, expanding the known classes of such sequences and providing evidence for their uniqueness.

Contribution

The paper presents two distinct methods to construct modular p-Stanley sequences for primes p>3, extending previous results and conjectures about their structure.

Findings

Two new constructions of modular p-Stanley sequences for p>3.

Identification of a larger class of integers n producing such sequences.

Numerical evidence supporting the conjecture that these are the only such sequences.

Abstract

Consider a set $A$ with no $p$-term arithmetic progressions for $p$ prime. The $p$-Stanley sequence of a set $A$ is generated by greedily adding successive integers that do not create a $p$-term arithmetic progression. For $p>3$ prime, we give two distinct constructions for $p$-Stanley sequences which have a regular structure and satisfy certain conditions in order to be modular $p$-Stanley sequences, a set of particularly nice sequences defined by Moy and Rolnick which always have a regular structure. Odlyzko and Stanley conjectured that the 3-Stanley sequence generated by $\{0,n\}$ only has a regular structure if $n=3^k$ or $n=2\cdot 3^k$. For $p>3$ we find a substantially larger class of integers $n$ such that the $p$-Stanley sequence generated from $\{0,n\}$ is a modular $p$-Stanley sequence and numerical evidence given by Moy and Rolnick suggests that these are the only $n$ for which the $p$-Stanley sequence generated by $\{0,n\}$ is a modular $p$-Stanley sequence. Our second class is a generalization of a construction of Rolnick for $p=3$ and is thematically similar to the analogous construction by Rolnick.