Simultaneous Small Fractional Parts of Polynomials

Abstract

Let $$f_1,\dots ,f_k\in \mathbb {R}[X]$$ f 1 , ⋯ , f k ∈ R [ X ] be polynomials of degree at most d with $$f_1(0)=\dots =f_k(0)=0$$ f 1 ( 0 ) = ⋯ = f k ( 0 ) = 0 . We show that there is an $$n<x$$ n < x such that $$\Vert f_i(n)\Vert _{\mathbb {R}/\mathbb {Z}}\ll x^{c/k}$$ ‖ f i ( n ) ‖ R / Z ≪ x c / k for all $$1\le i\le k$$ 1 ≤ i ≤ k for some constant $$c=c(d)$$ c = c ( d ) depending only on d. This is essentially optimal in the k-aspect, and improves on earlier results of Schmidt who showed the same result with $$c/k^2$$ c / k 2 in place of c/k.