# Search Results

## Abstract

For positive constants *a* > *b* > 0, let *P*
_{
T
} (*t*) denote the lattice point discrepancy of the body *tT*
_{
a,b
}, where *t* is a large real parameter and *T* = *T*
_{
a,b
} is bounded by the surface

*F*

_{ a,b }(

*t*) is an explicit continuous periodic function, and the remainder satisfies the (“pointwise”) estimate Δ

_{ T }(

*t*) ≪

*t*

^{11/8+ɛ }. Here it will be shown that this error term is only ≪

*t*

^{1+ɛ }

*in mean-square*, i.e., that

*ɛ*> 0.

## Abstract

This article provides an asymptotic formula for the number of integer points in the three-dimensional body

*a*>

*b*> 0 and large

*t*.

## Abstract

We improve the upper bound for the lattice point discrepancy of large spheres under conjectural properties of the real *L*-functions. In connection with this we give some new unconditional estimates for exponential and character sums of independent interest.

## Abstract

This paper provides estimates for exponential sums, combining classic tools of Van der Corput type with a deep result from the modern “discrete Hardy–Littlewood method”. As an application, an improved bound for the lattice point discrepancy of a large ellipsoid of rotation is deduced.

## Abstract

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let *α* be an arbitrary real root of a quadratic equation with integer coefficients; say,
. Given any rational number 0 < *x* < 1 (say, *x* = 1/2) and any positive integer *n*, we count the number of elements of the sequence *α*, 2*α*, 3*α*, ..., *nα* modulo 1 that fall into the subinterval [0, *x*]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected
number” *nx* from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ *n* ≤ *N*. Depending on *α* and *x*, we may need an extra additive correction of constant times logarithm of *N*; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm
of *N*. If *N* is large, the distribution of this renormalized counting number, as n runs in 1 ≤ *n* ≤ *N*, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as
*N* tends to infinity. This is the main result of the paper (see Theorem 1.1).

## Abstract

*α*be an arbitrary real root of a quadratic equation with integer coefficients; say,

*α*=

*x*< 1 (say,

*x*= 1/2) and any positive integer

*n*, we count the number of elements of the sequence

*α*, 2

*α*, 3

*α*, …,

*nα*modulo 1 that fall into the subinterval [0,

*x*]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number”

*nx*from the counting number, and study the typical fluctuation of this difference as

*n*runs in a long interval 1 ≤

*n*≤

*N*. Depending on

*α*and

*x*, we may need an extra additive correction of constant times logarithm of

*N*; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of

*N*. If

*N*is large, the distribution of this renormalized counting number, as

*n*runs in 1 ≤

*n*≤

*N*, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as

*N*tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on

*α*and

*x*, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums. The whole paper consists of an introduction and 17 sections. Part 1 contains the Introduction and Sections 1–7.

parentheses. Wald statistic cannot be used for testing the significance of sigma, p11, p21 and then stars for these parameters were omitted. Sources: LFS, CSO and own calculation. Fig. C1. Probability of state 1. Appendix D. Estimation of the “gap” equation