On $AP_3$ - covering sequences

TL;DR

The paper constructs a specific $AP_3$-covering sequence with a precise asymptotic density, improving previous bounds and demonstrating the existence of sequences with a limit superior of their normalized counting function equal to .

Contribution

It proves the existence of an $AP_3$-covering sequence with a sharper asymptotic density limit than previously known.

Findings

Existence of an $AP_3$-covering sequence with  as the limit superior.

Improvement over the previous upper bound of 34.

Provides a construction method for such sequences.

Abstract

Recently, motivated by Stanley sequences, Kiss, S\' andor and Yang introduced a new type sequence: a sequence $A$ of nonnegative integers is called an $AP_k$ - covering sequence if there exists an integer $n_0$ such that if $n > n_0$, then there exist $a_1\in A, \dots , a_{k-1}\in A$, $a_1<a_2<\cdots <a_{k-1}<n$ such that $a_1, \dots , a_{k-1}, n$ form a $k$-term arithmetic progression. They prove that there exists an $AP_3$ - covering sequence $A$ such that $\limsup\limits_{n\to\infty}{A(n)}/{\sqrt n}\le 34$. In this note, we prove that there exists an $AP_3$ - covering sequence $A$ such that $\limsup\limits_{n\to\infty}{A(n)}/{\sqrt n}=\sqrt{15}$.