Generalized Stanley sequences

Abstract: For an integer $k \ge 3$, let $A_{0} = \{a_{1}, \dots{} ,a_{t}\}$ be a set of nonnegative integers which does not contain an arithmetic progression of length $k$. The set $S(A)$ is defined by the following greedy algorithm. If $s \ge t$ and $a_{1}, \dots{} ,a_{s}$ have already been defined, then $a_{s+1}$ is the smallest integer $a > a_{s}$ such that $\{a_{1}, \dots{} ,a_{s}\} \cup \{a\}$ also does not contain a $k$-term arithmetic progression. The sequence $S(A)$ is called a \emph{Stanley sequence} of order $k$ generated by $A_{0}$. Starting out from a set of the form $A_{0} = \{0, t\}$, Richard P. Stanley and Odlyzko tried to generate arithmetic progression-free sets by using the greedy algorithm. In 1999, Erd\H{o}s, Lev, Rauzy, S\'andor and S\'ark\"ozy extended the notion of Stanley sequence to other initial sets $A_{0}$. In my talk I investigate some further generalizations of Stanley sequences and I give some density type results about them. This is a joint work with Csaba S\'andor and Quan-Hui Yang.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)