Simultaneous Small Fractional Parts of Polynomials

Given several real numbers α1,...,αk, how well can you simultaneously approximate all of them by rationals which each have the same square number as a denominator? Schmidt gave a clever iterative argument which showed that this can be done moderately well.

By using a general principle of 'little non-trivial additive structure in rationals' and some ideas from additive combinatorics and the geometry of numbers, I'll describe how this can be improved to give a close-to-optimal answer when k is large.