Quadratic Weyl sums, automorphic functions and invariance principles

Abstract

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution.

Autocorrelation functions for quantum particles in supersymmetric Pöschl-Teller potentials

“…In this section we describe explicitly the random variable X featured in Theorem 1.1. We refer to §2 of [1] for more details. 2020/10/26 14:18 6 F. Cellarosi…”

“…Let ω be the standard symplectic form on R 2 , ω(ξ, ξ ) = x y − yx , where ξ = x y , ξ = x y . The Heisenberg group H(R) is defined as R 2 × R with the multiplication law (ξ, t)(ξ , t ) = ξ + ξ , t + t + 1 2 ω(ξ, ξ ) . (2.8)…”

“…is a random variable whose law is simply the push forward of the normalized Haar measure 3 π 2 µ onto C 2 via X. The properties of the law of each component Θ f (Γg) when Γg is Haar-random on Γ\G have been studied in [1]. In particular, we have the following Lemma 2.2 ([1]) Let η > 1 and f ∈ S η .…”

Cornu Spirals and the Triangular Lacunary Trigonometric System

“…More pertinent to the current work, Lehmer showed that Cornu spirals arise from incomplete Gaussian summations [2], which encompass the triangular lacunary trigonometric system of the current work. Berry and Goldberg developed a renormalization procedure for such Cornu spirals [4] as did Sinai [31] and Fedotov and Klopp [32], while Cellarosi and Marklof investigated the related quadratic Weyl summations [33], and Paris provided much insight into various expansions of such systems along with asymptotic behavior [3]. Indeed families of, at times elaborate, combinations of Cornu spirals arise when considering F n,q .…”

Pair correlation for quadratic polynomials mod 1

“…It is well-known that the operators U φ : f → f φ are unitary; note that f π/2 =f . Moreover, Θ f is a smooth function on G (see [5] for example). LetΓ be the subgroup of G defined bỹ…”