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Journal ArticleDOI

Exponential sums and lattice points III

01 Mar 1993-Proceedings of The London Mathematical Society (Oxford University Press)-Vol. 87, Iss: 3, pp 591-609
TL;DR: In the case of the Dirichlet divisor problem, the number of points of the integer lattice in a planar domain bounded by a piecewise smooth curve has been shown to be upper bounded by the radius of the maximum radius of curvature as mentioned in this paper.
Abstract: The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri?Iwaniec?Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri?Iwaniec?Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.
Citations
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Journal ArticleDOI
TL;DR: It is found that there is an inherent limitation in the precision of computing the Zernike moments due to the geometric nature of a circular domain.
Abstract: We give a detailed analysis of the accuracy of Zernike moments in terms of their discretization errors and the reconstruction power. It is found that there is an inherent limitation in the precision of computing the Zernike moments due to the geometric nature of a circular domain. This is explained by relating the accuracy issue to a celebrated problem in analytic number theory of evaluating the lattice points within a circle.

276 citations


Cites background from "Exponential sums and lattice points..."

  • ...Nevertheless, the quantity G(n) has been extensively examined in the analytic number theory with the relation to the so-called lattice points of a circle problem due originally to Gauss [ 5 ], [6]....

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  • ...See [ 5 ], [6], and the references cited therein....

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  • ...See also [ 5 ] for a slightly sharper result, equivalent to the latter one up to a logarithmic factor....

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Journal ArticleDOI
TL;DR: Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids as mentioned in this paper.

254 citations

Book
01 Jan 2004
TL;DR: Questions of particular interest include how images and image subsets are digitized; how geometric properties are defined for digitized sets; the computational complexity of computing them--in particular, whether they can be computed using simple (e.g., local) operations; characterizing image operations that preserve them; and characterizing digital objects that could be the digitizations of real objects that have given geometric properties.
Abstract: Digital geometry is the study of geometrical properties of subsets of digital images. If the digitization is sufficiently fine-grained, such properties can be regarded as approximations to the corresponding properties of the "real" sets that gave rise, by digitization, to the digital sets; but it is also important to define how the properties can be computed for the digital sets themselves. Questions of particular interest include how images and image subsets are digitized; how geometric properties are defined for digitized sets; the computational complexity of computing them--in particular, whether they can be computed using simple (e.g., local) operations; characterizing image operations that preserve them; and characterizing digital objects that could be the digitizations of real objects that have given geometric properties. Concepts that have been extensively studied include topological properties (connected components, boundaries); curves and surfaces; straightness, curvature, convexity, and elongatedness; distance, extent, length, area, surface area, volume, and moments; shape description, similarity, symmetry, and relative position; shape simplification and skeletonization.

249 citations

Book
22 Aug 1996
TL;DR: In this paper, a simple exponential sum with modular form coefficients was proposed for the Riemann zeta function. But the first and second conditions of the exponential sum were not considered.
Abstract: Introduction Part I Elementary Methods 1 The rational line 2 Polygons and area 3 Integer points close to a curve 4 Rational points close to a curve Part II The Bombieri-Iwaniec Method 5 Analytic methods 7 The simple exponential sum 8 Exponential sums with a difference 9 Exponential sums with a difference 10 Exponential sums with modular form coefficients Part III The First Spacing Problem: Integer Vectors 11 The ruled surface method 12 The Hardy Littlewood method 13 The first spacing problem for the double sum Part IV The Second Spacing Problem: Rational vectors 14 The first and second conditions 15 Consecutive minor arcs Part V Results and Applications 17 Exponential sum theorems 18 Lattice points and area 19 Further results 20 Sums with modular form coefficients m 21 Applications to the Riemann zeta function 22 An application to number theory: prime integer points Part IV Related Work and Further Ideas 23 Related work 24 Further ideas References

241 citations

Journal ArticleDOI
TL;DR: Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids.
Abstract: Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long-wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids. All perfect crystals, perfect quasicrystals and special disordered systems are hyperuniform. Thus, the hyperuniformity concept enables a unified framework to classify and structurally characterize crystals, quasicrystals and the exotic disordered varieties. While disordered hyperuniform systems were largely unknown in the scientific community over a decade ago, now there is a realization that such systems arise in a host of contexts across the physical, materials, chemical, mathematical, engineering, and biological sciences, including disordered ground states, glass formation, jamming, Coulomb systems, spin systems, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, number theory, stochastic point processes, integral and stochastic geometry, the immune system, and photoreceptor cells. Such unusual amorphous states can be obtained via equilibrium or nonequilibrium routes, and come in both quantum-mechanical and classical varieties. The connections of hyperuniform states of matter to many different areas of fundamental science appear to be profound and yet our theoretical understanding of these unusual systems is only in its infancy. The purpose of this review article is to introduce the reader to the theoretical foundations of hyperuniform ordered and disordered systems. Special focus will be placed on fundamental and practical aspects of the disordered kinds, including our current state of knowledge of these exotic amorphous systems as well as their formation and novel physical properties.

228 citations


Additional excerpts

  • ...[370] and references therein....

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References
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MonographDOI
25 Jan 1991
TL;DR: The van der Corput method can be applied to problems such as upper bounds for the Riemann-Zeta function, the Dirichlet divisor problem, the distribution of square free numbers and the Piatetski-Shapiro prime number theorem.
Abstract: This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums These arise in many problems in analytic number theory It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem

470 citations

Journal ArticleDOI
TL;DR: In this article, an axiomatised version of the Bombieri?Iwaniec method has been proposed, where an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for $S$.
Abstract: A Van der Corput exponential sum is $S = \Sigma \exp (2 \pi i f(m))$, where $m$ has size $M$, the function $f(x)$ has size $T$ and $\alpha = (\log M) / \log T < 1$. There are different bounds for $S$ in different ranges for $\alpha $. In the middle range where $\alpha $ is near ${1\over 2}$, $S = O(\sqrt{M} T^{\theta + \epsilon })$. This $\theta $ bounds the exponent of growth of the Riemann zeta function on its critical line ${\rm Re} s = {1\over 2}$. Van der Corput used an iteration which changed $\alpha$ at each step. The Bombieri?Iwaniec method, whilst still based on mean squares, introduces number-theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of $y = f'(x)$ coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1?40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some ?resonance curve?, one of a family of curves in some dual space, now calculated accurately in the paper ?Resonance curves in the Bombieri?Iwaniec method?, which is to appear in Funct. Approx. Comment. Math. We turn the whole Bombieri?Iwaniec method into an axiomatised step: an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for $S$. Ends and cusps of resonance curves are treated separately. Bounds for sums of type $S$ lead to bounds for integer points close to curves, and another branching iteration. Luckily Swinnerton-Dyer's method is stronger. We improve $\theta $ from 0.156140... in the previous paper and monograph to 0.156098.... In fact $(32/205 + \epsilon , 269/410 + \epsilon)$ is an exponent pair for every $\epsilon > 0$.

207 citations