On a Problem of Arnold on Uniform Distribution

In this note we consider some quantitative versions of conjectures made by Arnold related to Galois dynamics in finite fields. We refine some results by Shparlinski using exponential sum results. The present note is a refinement of work of I. Shparlinski [S] on the ergodic properties of certain dynamical systems associated to multiplication in finite fields and originating from some problems posed by I. V. Arnold (see [A]). As shown in [S] the issue (explained below) turns out to be ultimately connected to questions on incomplete exponential sums of Gauss type. Using the ‘standard’ bounds on such sums, (combined with discrepancy estimates) Arnold’s question was settled affirmatively in [S]. In fact, the result obtained in [S] shows uniform distribution of even much shorter orbits than considered in [A] and raises the natural question: what is the true condition to establish this phenomenon? Our first and main aim here is to show how recently obtained exponential sum bounds in fields Fp (see [BC]) and going well beyond Gauss’ estimates, permit to prove uniform distribution of orbits of length M ≥ p for any fixed δ > 0, while in [S] the condition M À p(log p) is required. Next, a selfcontained account is given of how to derive directly from the exponential sum bound the discrepancy estimates for smooth domains (without first ‘passing through boxes’). No effort has been made however to optimize the error term. Next, we describe our problem in details. Let p be a large prime, and let n ∈ Z be fixed. A finite field Fpn ∼= Fp[ξ] can be viewed as a n-dimensional vector space over Fp via the correspondence n−1 ∑ j=0 xjξ j ←→ (x0, . . . , xn−1). Let am = (am,0, . . . , am,n−1) be the vector corresponding to ξ = ∑n−1 j=0 am,j ξ j . After identifying Fp with {0, 1, . . . , p− 1}, we have 1 p am ∈ [0, 1] ⊂ Q, where m = 1, . . . ,M. Typeset by AMS-TEX 1 We are interested in the distribution of { 1 p am : m = 1, . . . ,M} in [0, 1]. For example, if M = p − 1, we get regular lattice with only one point (0, . . . , 0) missing. Arnold Conjectured [A] that there is uniform distribution even for small values of M . More precisely, let Ω ⊂ [0, 1] be a region with smooth boundary and denote Nξ(M, Ω) = ∣{1 ≤ m ≤ M : 1 p am ∈ Ω} ∣. (1) Conjecture. (Arnold) Nξ(M, Ω) = M vol Ω + o(M) even for small values of M . By ‘small’ Arnold refers to M = o(p). Shparlinski proved that M À pn2 (log p). In this note we will improve the lower bound on M obtained by Shparlinski. Theorem 1. Let Nξ(M, Ω) be defined as in (1). Then Nξ(M, Ω) = M vol Ω + o(M) for M > p for any fixed δ > 0. In [S], the author also considers the following general problem. Let f(x) ∈ Z[x] be a fixed nonconstant polynomial and write