Proposition 1. Let be an irrational number and denote the set of fractional parts of for all . Then is dense in .
Proof. We shall first prove that is dense at , then proceed to the whole range of . Without loss of generality, is assumed to be within .
To show that is dense at , we claim that there exists a such that . Denote the largest integer satisfying , then . If , then is what we need, else let . Denote the largest integer satisfying , then and , that is, is what we need. Then apply this procedure to and so on, which leads to a sequence approaching to at least as fast as .
Further, for any , we can find a such that by the above argument. Then for any , , thus is dense at . This completes the proof.
Alternatively, we can prove Prop. 1 by the Dirichlet’s approximation theorem, which states that for any real number and positive integer , there exit integers and such that and . Thus is dense at and then dense everywhere on .
Remark 1. Prop. 1 can be formulated on the unit circle , that is, the set for any irrational number is dense on .
Remark 2. The interval can be replaced with where as long as is irrational ( might be rational).
Remark 3. If we replace with its subset , where , will still be dense in . This is because , in which is dense in and amounts to a constant (circular) shifting. On the other hand, for any infinite subset , will be dense somewhere in , since is compact, but is not necessarily dense everywhere in , for example, .
Problem 1. Let be the largest interval between two adjacent points from on . By Prop. 1, monotonically goes to . But what is the speed of decaying of as goes to infinity? For example, could it be ?
Problem 2. Let be an irrational number. Then, for what kind of , is dense in ? Equivalently, we could ask for what kind of subset such that the set is dense in . Specifically, is dense on ? Let be the set of prime numbers, is dense on ?