Seminar series
Date
Thu, 19 May 2016
16:00
16:00
Location
L6
Speaker
Roger Baker
Organisation
Brigham Young University
For $k \geq 3$ we give new values of $\rho_k$ such that
$$ \| \alpha p^k + \beta \| < p^{-\rho_k} $$
has infinitely many solutions in primes whenever $\alpha$ is irrational and $\beta$ is real. The mean
value results of Bourgain, Demeter, and Guth are useful for $k \geq 6$; for all $k$, the results also
depend on bounding the number of solutions of a congruence of the form
$$ \left\| \frac{sy^k}{q} \right\| < \frac{1}{Z} \ \ (1 \leq y \leq Y < q) $$
where $q$ is a given large natural number.