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12 votes
1 answer
617 views

Convergence of the series involving Mobius functions $\sum_{k,d} \mu(d) x_{kd}$

(I originally asked this question here, but the problem appears much more difficult than I think after a moment of thought, so I think it might be more suitable to post it here. Please tell me if this ...
Ma Joad's user avatar
  • 1,755
3 votes
1 answer
332 views

Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part. Let $a < b$ be two numbers in $[0, 1]$. Then by ...
Mohammad Ali Karami's user avatar
13 votes
1 answer
1k views

Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$ runs over the integers? The existence of the limes inferior follows from Dirichlet's approximation theorem, but the ...
Y.X's user avatar
  • 399