Breadcrumb
Diophantine approximation with primes and almost primes
Wed 30 January 2013, 11:00
Alastair Irving
Oxford
Organiser: Tim Browning
ABSTRACT
Vinogradov showed that for any irrational \alpha and any \tau<1/5 there are infinitely many primes p for which
\|p\alpha\|\leq p^{-\tau}.
The exponent 1/5 was improved by several authors, culminating in the result of Matomaki that we can take \tau<1/3. The key ingredient in these results is estimates for certain "Type I" and "Type II" sums. I will describe the existing estimates for such sums. For \tau\geq 1/3 I will show how some Type II sums may be estimated. These results are not sufficient to improve the exponent for primes but they give a new result concerning Diophantine Approximation with numbers having precisely two prime factors.