On the Generalized Antiaverage Problem

Abstract

For integers $k,\theta \le 3$ and $\beta \ge 1$, an integer $k$-set $S$ with the smallest element $0$ is a $(k;\beta ,\theta )$-free set if it does not contain distinct elements $a_{i,j}$ ($1\le i\le j\le \theta$) such that $\sum _{j=1}^{\theta -1}{a}_{i,j}=\beta a_{i_{\theta }}$. The largest integer of $S$ is denoted by $max(S)$. The generalized antiaverage number $\lambda (k;\beta ,\theta )$ is equal to $min\{max(S):S\text{ is a }(k^0;\delta ,0)\text{-free set}\}$. We obtain:(1) If $\beta \notin \{\theta -2,\theta -1,\theta \}$, then $\lambda (m;\beta ,\theta )\le (\theta -1)(m-2)+1$; (2) If $\beta \ge \theta -1$, then $\lambda (k;\beta ,\theta )\le \underset{k=m+n}{min}\{\lambda (m;\beta ,\theta )+\beta \lambda (n;\beta ,\theta )+1\}$, where $k=m+n$ with $n>m\ge 3$ and $\lambda (2n;\beta ,\theta )\le \lambda (n;\beta ,\theta )(\beta +1)+\epsilon$, for $\epsilon =1$ for $\theta =3$ and $\epsilon =0$ otherwise.