On the distribution of $\alpha p+\beta$ modulo one for primes of type $p=a{r}^2+1$

A classical problem in analytic number theory is to study the distribution of fractional part $\alpha p+\beta$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p=a{r}^2+1$ and prove that its distribution has a similar property.