Geodesic
Maximal upper asymptotic density of sets of integers with missing differences from a given set
Mathematica Bohemica, Tome 140 (2015) no. 1, pp. 53-69

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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
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 $a$ and $\gcd (a,b)=1.$
DOI : 10.21136/MB.2015.144179
Classification : 11B05
Keywords: upper asymptotic density; maximal density 
Pandey, Ram Krishna. Maximal upper asymptotic density of sets of integers with missing differences from a given set. Mathematica Bohemica, Tome 140 (2015) no. 1, pp. 53-69. doi: 10.21136/MB.2015.144179
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