Let $p\_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p\_n \alpha\}$ is equidistributed. Is it known whether the sequence $\{p\_n^2 \alpha \}$ is equidistributed?



Equidistribution of $\{p_n^2α\}$
Yes - this follows from a general theorem of Bergelson, Kolesnik, Madritsch, Son, and Tichy (Theorem 2.1 in <https://people.math.osu.edu/bergelson.1/BKMS_PrimePowers.pdf>):


Let $\xi(x)=\sum\_{j=1}^m\alpha\_j x^{\theta\_j}$ be a polynomial with real coefficients $\alpha\_i\in\mathbb{R}$ such that $0<\theta\_1<\cdots <\theta\_m$ and either


a) at least one of the $\theta\_j$ is not at integer, or


b) at least one of the $\alpha\_j$ is irrational.


If $\xi$ satisfies at least one of these conditions then $(\xi(p))$ is uniformly distributed. 


Your special case $\xi(x)=\alpha x^2$ was probably known earlier, but I can't find a reference for that at the moment.


EDIT: This theorem in the case $\theta\_j$ all integers was proved by Rhin in 1973 using Vinogradov's method, see <https://mathscinet.ams.org/mathscinet-getitem?mr=323731>. According to that Mathscinet review, this result was also implicit in Vinogradov's book.


