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Suppose $k+1$ runners having nonzero distinct constant speeds run laps on a unit-length circular track. The Lonely Runner Conjecture states that there is a time at which a given runner is at distance at least $1/(k+1)$ from all the others. The conjecture has been already settled up to seven ($k\le 6$) runners while it is open for eight or more runners. In this paper the conjecture has been verified for four or more runners having some particular speeds using elementary tools.

Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\overline{\delta }\left(S\right)$ denote the upper asymptotic density of $S$. We consider the problem of finding $$\mu \left(M\right):=\underset{S}{sup}\overline{\delta }\left(S\right),$$ 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 $\mu \left(M\right)$ where $M=\{a,b,a+nb\}$ for all even integers and for all sufficiently large odd integers $n$ with $a<b$ and $gcd(a,b)=1.$