Ram Krishna Pandey, Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-Haridwar Highway, Roorkee 247667, Uttarakhand, India, e-mail: ramkpfma@iitr.ac.in
Abstract: 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 $\gcd(a,b)=1.$
Keywords: upper asymptotic density; maximal density
Classification (MSC 2010): 11B05
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