Event description
The classical Dirichlet approximation theorem, whose standard proof is a simple application of the Pigeonhole Principle, says that given any irrational number $\alpha$, there are infinitely many rational numbers $a/p$ (with $\gcd(a,p)=1,p\ge 1$) such that $|\alpha-a/p|\le 1/p^2$. Here, $p$ is an arbitrary positive integer. If one restricts the allowed denominators $p$ to be rational primes, the result is known as Vinogradov's theorem. What if one further restricts the primes that are allowed in the denominator? For example, suppose you allow only primes of the form $p=x^2+2y^2$? Or only primes which can be written in the form $p= x^3+2y^3+4z^3-6xyz$? Can we still assure the existence of infinitely many such approximations? The answer is yes, but the approximation may no longer be as accurate as $1/p^2$. We shall outline a proof of the relevant result in this talk.