There is the statement of Van der carport theorem:
Given a sequences in , if , is uniformly distributed, then is uniformly distributed.
I do not know how to establish this theorem with no extra condition, but this result is true at least for polynomial flow.
This type of trick could also establish the following result, which could be understand as a discretization of the Vinegradov lemma.
Uniformly distribution result of : Given , coverages to a uniformly distribution in as .
This trick could also help to establish estimate of correlation of low complexity sequences and multiplicative function, such as result:
Maybe with the help of B-Z-S theorem.
The standard estimate of Mobius function is:
The error term is true for .