Dense Sequence

[Navid]

Can anyone guide me, where i can find an article about the density proof of some sequence like $n^{2} a-\lfloor n^{2} a\rfloor$ and $2^{n} a-\lfloor 2^{n} a\rfloor$ where $a$ is an irrational number?

[g______d]

Navid в сообщении #1150665 писал(а):

$n^{2} a-\lfloor n^{2} a\rfloor$

Follows from van der Corput's theorem.

[Navid]

Thank you!, I know about van der Corput's Uniform Distribution theorem and Herman Weyl's lemma which proves that the former sequence is indeed U.D, But i wonder, whether there are another proof for its denseness and also the second sequence, Did you remember any helpful Article?

[g______d]

Navid в сообщении #1150706 писал(а):

also the second sequence

It's against the rules to give complete solutions here. I can only give you a hint: look at base 2 expansion of $a$. You can easily construct a lot of examples of irrational $a$ whose trajectories are not dense.

[Deggial]

Тема перемещена из форума «Олимпиадные задачи (М)» в форум «Помогите решить / разобраться (М)»
Причина переноса: в соответствующий раздел