The theorem of Bourgain, Demeter and Guth on Vinogradov's mean value has an application to the minimal fractional part of a polynomial of degree k without constant term, evaluated at 1,...,N. The Heilbronn-Danicic method is superseded for k > 5 in the case of a monomial, but there is an unfortunate loss of a factor of 2 in the exponent when we replace a monomial by a general polynomial.
In this talk we describe joint work with Trevor Wooley in which this loss is partially regained. There is a strong connection with work of Bob Vaughan on approximation to Weyl sums.
I discuss briefly an application of this work to fractional parts of polynomials over the primes.