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The mean lattice point discrepancy

Published online by Cambridge University Press:  20 January 2009

M. N. Huxley
M. N. Huxley
Affiliation:
School of Mathematics, University of Wales College of Cardiff, 23, Senghenydd Road, Cardiff CF2 4YH
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Abstract

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Consider a sufficiently smooth simple closed convex plane curve enclosing the origin, expanding linearly with time. The root mean square of the discrepancy (number of lattice points minus area) from time t = M to t = M + 1 is almost as small as the root mean square discrepancy from time t = 0 to t = M, so the discrepancy has no memory.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 3 , October 1995 , pp. 523 - 531
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Chaix, H., Demonstration elementaire d'un theoreme van der Corput, C. R. Acad. Sci. Paris 275 (1972), A883–885.Google Scholar
2.Colin De Verdiere, Y.Nombre de points entiers dans une famille homothetique de domains de R, Ann. Sci. EC. Norm. Sup. (4) 10 (1977), 559–576.CrossRefGoogle Scholar
3.Van Der Corput, J. G., Uber Gitterpunktc in der Ebene, Math. Ann. 81 (1920), 1–20.CrossRefGoogle Scholar
4.Herz, C. S., Fourier transforms related to convex sets, Ann. of Math. 75 (1962), 81–92.CrossRefGoogle Scholar
5.Herz, C. S., On the number of lattice points in a convex set, Amer. J. Math. 84 (1962), 126–132.CrossRefGoogle Scholar
6.Hlawka, E., Integrale auf konvexen Korpern, Monatsh. Math. 54 (1950), 1–36, 81–99.CrossRefGoogle Scholar
7.Huxley, M. N., The rational points close to a curve, Ann. Scuola Norm. Sup. Pisa (Cl. Sci.), (4) 21 (1994), 357–375.Google Scholar
8.Huxley, M. N., Area, Lattice Points, and Exponential Sums (London Math. Soc.), to appear.Google Scholar
9.Kendall, D. G., On the number of lattice points inside a random oval, Quarterly J. Math. (Oxford) 19 (1948), 1–26.Google Scholar
10.Nowak, W. G., On the average order of the lattice rest of a convex planar domain, Math. Proc. Cambridge Philos. Soc. 98 (1985), 1–4.CrossRefGoogle Scholar
11.Nowak, W. G., An Q-estimate for the lattice rest of a convex planar domain, Proc. Royal Soc. Edinburgh 100A (1985), 295–299.CrossRefGoogle Scholar
12.Sierpinski, W., Sur un probleme du calcul des fonctions asymptotiques, Prace Mat.-Fiz. 17 (1906), 77–118.Google Scholar
13.Voronoi, G., Sur une fonction transcendente et ses applications a la sommation de quelques series, Ann. Ecole Norm. Sup (3) 21 (1904), 207–267, 459–533.CrossRefGoogle Scholar