Relative dense and dense

Question

A set $A\subset\mathbb{R}$ is called relatively dense if there exists a $M>0$, s.t. every interval of length $M$ contains a $x\in A$.

I cannot find in the literature any connections to a dense subset. Is it true that any dense set is a relative dense set? The other implication is clearly not true.

See epsilon-dense property AND $\epsilon$- dense subsets AND $\epsilon$-dense subsets on $\mathbb R/\mathbb Z$ AND this Wikipedia article. — Dave L. Renfro, May 30 '18 at 18:55
Does this thin definition have a use? It is hard to imagine anything dense about Z or nZ for huge n. — William Elliot, May 30 '18 at 19:22
It actually has in the theory of almost sure periodic functions. — RandomUser, May 31 '18 at 18:23

score 2 · Answer 1 · answered May 30 '18 at 18:25

2

If $A$ is dense, then, for every $M>0$, every interval with length $M$ contains an element of $A$. Therefore, $A$ is relatively dense.

answered May 30 '18 at 18:25

José Carlos Santos

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Relative dense and dense

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