Distribution of the error term for the number of lattice points inside a shifted circle

Abstract

We investigate the fluctuations in N a (R), the number of lattice points n £ Z 2 inside a circle of radius R centered at a fixed point α e [0, I) 2 . Assuming that R is smoothly (e.g., uniformly) distributed on a segment 0 ^ R ^ T 9 we prove that the random variable ---7= -has a limit distribution as T -> oo (independent of the distribution of R\ which is absolutely continuous with respect to Lebesgue measure. The density p u (x) is an entire function of x which decays, for real x, faster than exρ ( -|x| 4~ε ). We also obtain a lower bound on the distribution function P α (x) = J^ p Λ (y)dy which shows that P α (x) and 1 -P Λ (x) decay when x -> oo not faster than exp ( -x 4+ε ). Numerical studies show that the profile of the density p α (x) can be very different for different α. For instance, it can be both unimodal and bimodal. We show that J^ xp α (x)dx = 0, and the variance D α = J^ x 2 p (X (x)dx depends continuously on α. However, the partial derivatives of D a are infinite at every rational point α e Q 2 , so D α is analytic nowhere. ContentsI. Introduction 434 II. Ergodic Theorem 443 III. Almost Periodicity of the Error Function 445 IV. Upper Bound on the Error Term Distribution Density 450 V. Lower Bound on the Error Term Distribution Function 454 Appendix A. Proof of Theorem 4.1 456 Appendix B. Proof of Theorem 4.3 460 References 468

WKB to all orders and the accuracy of the semiclassical quantization

“…This is of course compatible with the very important semiclassical theory of delta statistics ∆(L) (spectral rigidity) by Berry (1985), employing the Gutzwiller periodic orbit theory (1990), where agreement with predictions of random matrix theories and with the experimental and numerical data has been obtained at large L. Also, Berry and Tabor (1977) have used torus quantization of integrable systems (with many degrres of freedom), predicting the Poissonian (exponential) energy level distribution. Our results show that their result cannot be rigorous, especially as we know some counterexamples of integrable systems with non-Poissonian statistics (Bleher et al 1993), and also know that their approximation does not take into account the nonperturbative tunneling effects, but it is nevertheless a heuristic argument explaining why typically we do observe Poissonian statistics in classically integrable systems. By typically we mean that the set of exceptions has a small or even vanishing measure.…”

Mean square value of exponential sums related to the representation of integers as sums of squares

“…, m k ). The above sums were studied by Bleher, Cheng, Dyson and Lebowitz [2], Bleher and Dyson [3]- [5] and Bleher and Bourgain [1] in connection with the fluctuations of the number of lattice points inside a large sphere centered at α.…”

Transition of Spectral Statistics of a Spin-1/2 Coupled Quartic Oscillators from GSE to Integrable Regime

“…In term of the random matrix theory, it is proven that the MFD becomes Gaussian in the limit of infinite rank, 10) if the system is chaotic. Indeed, in our work, the MFD of the chaotic regimes always becomes Gaussian, irrespective of the kinds of the Gaussian ensemble of our chaotic limits.…”