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Keywords:
upper asymptotic density; maximal density
upper asymptotic density; maximal density
Summary:
Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\overline \delta (S)$ denote the upper asymptotic density of $S$. We consider the problem of finding \[\mu (M):=\sup _{S}\overline \delta (S),\] where the supremum is taken over all sets $S$ satisfying that for each $a,b\in S$, $a-b \notin M.$ In this paper we discuss the values and bounds of $\mu (M)$ where $M = \{a,b,a+nb\}$ for all even integers and for all sufficiently large odd integers $n$ with $a<b$ and $\gcd (a,b)=1.$
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