A note on character sums over short moving intervals

Abstract

We investigate the sums (1/ √ H) X<n≤X+H χ(n), where χ is a fixed non-principal Dirichlet character modulo a prime q, and 0 ≤ X ≤ q − 1 is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided H → ∞ and (log H)/ log q → 0 as q → ∞, and Lamzouri conjectured these should hold subject to the much weaker upper bound H = o(q/ log q). We prove this is false for some χ, even when H = q/ log A q for any fixed A > 0. On the other hand, we show it is true for "almost all" characters on the range q 1−o(1) ≤ H = o(q). Using Pólya's Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.Conjecture 1 (Lamzouri, 2013). Suppose that H → ∞ but H = o(q/ log q) as the prime q → ∞. Then if χ = • q is the Legendre symbol, and if X ∈ {0, 1, ..., q − 1} is

Paucity phenomena for polynomial products

“…Thus, there is no paucity phenomenon in the linear case and our mild condition is necessary. However, we remark that in the companion paper [19] of the authors and Pandey, the analogous counting problem is solved for 𝑃(𝑥) = 𝑥 with short interval support [𝑁, 𝑁 + 𝐻], 𝐻 ⩽ 𝑁∕(log 𝑁) 𝐶𝑘 2 , 1 and this is one of the key steps in making progress toward a recent question of Harper [10]. The "paucity" in this case comes from the "shortness" of the interval; in comparison, the "paucity" in the present paper comes from the fact that 𝑃(𝑥) is "genuinely of degree ⩾ 2" (i.e., it has at least two distinct complex roots).…”

The limiting distribution of Legendre paths

“…, p − 1}, the short sum S p (x + H) − S p (x) tends to a normal distribution with mean zero and variance H, provided log H = o(log p) and H → ∞ as p → ∞. Recently, Harper [12] showed that this is no longer the case if H is much larger, namely when H ⩾ p/(log p) A , and A > 0 is a fixed constant.…”

Paucity phenomena for polynomial products

“…However, we remark that in the companion paper [15] of the authors and Pandey, the analogous counting problem is solved for P (x) = x with short interval support [N, N +H], H ≤ N/(log N) Ck 2 , 1 and this is one of the key steps in making progress towards a recent question of Harper [7]. The "paucity" in the linear case comes from the "shortness" of the interval; in comparison, the "paucity" in the present paper comes from the fact that P (x) is "genuinely of degree ≥ 2" (i.e.…”