Simultaneous small fractional parts of polynomials

Abstract

Let f 1 , . . . , f k ∈ R[X] be polynomials of degree at most d with f 1 (0) = • • • = f k (0) = 0. We show that there is an n < x such that f i (n) R/Z ≪ x c/k for all 1 ≤ i ≤ k for some constant 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 in place offor all i ∈ {1, . . . , k}.Choosingin Theorem 1.1 gives the improvement mentioned above. In the language of [14], this confirms the conjecture that an arbitrary system of polynomials withThen there is a positive integer n < x such thatfor all i ∈ {1, . . . , k}.Here c d > 0 is a constant depending only on d, and the implied constant depends only on d and k.As with previous works, a noteworthy feature of Theorem 1.1 and Corollary 1.2 is that the result is completely uniform over the coefficients of the polynomials with the implied constants depending only on d and k.The proof as given in this paper would yield a constant c d in Corollary 1.2 which is exponentially small in d (c d = 10 −d would probably suffice), but it is likely that with only a small amount of additional effort the constant could be taken to be of the form c d = C/d 2 or perhaps even C/(d + d 2 /k) for a relatively small explicit absolute constant C. In the interests of emphasizing the main ideas we have chosen not to pursue such explicit bounds in the d-aspect. Similarly we have made no effort to control the implied constant's dependence on d or k, although it is likely that adapting the ideas behind [11, Proposition A.2] would give a reasonable and explicit dependence on k and d.