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# A mean-square bound concerning the lattice discrepancy of a torus in ℝ3

Acta Mathematica Hungarica
Authors:
V. Garcia
and
W. Nowak

## 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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\partial \tau _{a,b} :\left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {(a + b\cos \alpha )\cos \beta } \\ {(a + b\cos \alpha )\sin \beta } \\ {b\sin \alpha } \\ \end{array} } \right), 0 \leqq \alpha ,\beta < 2\pi .$$ \end{document}
In a previous paper  it has been proved that
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_\tau (t) = \mathcal{F}_{a,b} (t)t^{3/2} + \Delta _\tau (t),$$ \end{document}
where 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
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_0^T {(\Delta _\tau (t))^2 dt} \ll T^{3 + \varepsilon }$$ \end{document}
for any ɛ > 0.

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# The lattice point discrepancy of a torus in ℝ3

Acta Mathematica Hungarica
Author:
W. Nowak

## Abstract

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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ \end{gathered} \right) = t\left( \begin{gathered} (a + r\cos \alpha )\cos \beta \hfill \\ (a + r\cos \alpha )\sin \beta \hfill \\ r\sin \alpha \hfill \\ \end{gathered} \right),0 \leqq \alpha ,\beta < 2\pi ,0 \leqq r \leqq b,$$ \end{document}
for fixed a > b > 0 and large t.

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# The sphere problem and the L-functions

Acta Mathematica Hungarica
Authors:
Fernando Chamizo
and
Elena Cristóbal

## 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.

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# Higher order derivative tests for exponential sums incorporating the Discrete Hardy–Littlewood method

Acta Mathematica Hungarica
Author:
Werner Georg Nowak

## 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.

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# Randomness of the square root of 2 and the giant leap, part 2

Periodica Mathematica Hungarica
Author:
József Beck

## 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, 2α, 3α, ..., 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 ≤ nN. 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 ≤ nN, 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).

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# Randomness of the square root of 2 and the Giant Leap, Part 1

Periodica Mathematica Hungarica
Author:
József Beck

## 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, α =
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sqrt 2$$ \end{document}
. 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, 2α, 3α, …, 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 ≤ nN. 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 ≤ nN, 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.
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# The output employment elasticity and the increased use of temporary contracts: Evidence from Poland

Acta Oeconomica
Authors:
Kristof Bartosik
and
Jan Mycielski

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

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# Expansionary or restrictive policies in the Eurozone? Dominating trends in the first two decades

Acta Oeconomica
Authors:
Joanna Stawska
and
Paulo Jorge Reis Mourao

p 21

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