Random fluctuations of convex domains

and lattice points

Authors:
Alex Iosevich and Kimberly K. J. Kinateder

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2981-2985

MSC (1991):
Primary 42Bxx; Secondary 60G99

DOI:
https://doi.org/10.1090/S0002-9939-99-04879-0

Published electronically:
April 27, 1999

MathSciNet review:
1605972

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we examine a random version of the lattice point problem. Let denote the class of all homogeneous functions in of degree one, positive away from the origin. Let be a random element of , defined on probability space , and define

for . We prove that, if , where , then

where , the expected volume. That is, on average, . We give explicit examples in which the Gaussian curvature of is small with high probability, and the estimate holds nevertheless.

**[vdC21]**J. G. van der Corput,*Zahlentheoretische Abschatzungen*, Math. Ann.**84**(1921).**[Dur91]**R. Durrett,*Probability: Theory and Examples*, Wadsworth and Brooks-Cole (1991). MR**91m:60002****[Hlw50]**E. Hlawka,*Uber Integrale auf konvexen Korper I*, Monatsh Math**54**(1950).**[Hux97]**M. Huxley,*Area, Lattice Points, and Exponential Sums*, London Math. Soc. Mon. New Series**13**(1996). MR**97g:11088****[KS91]**I. Karatzas and S. E. Shreve,*Brownian Motion and Stochastic Calculus*, Springer -Verlag (1991). MR**92h:60127****[Rand69]**B. Randol,*On the asymptotic behavior of the Fourier transform of the indicator function of a convex set*, Trans. of the A.M.S.**139**(1969), 279-285. MR**40:4678b****[So93]**C. D. Sogge,*Fourier Integrals in Classical Analysis*, Cambridge Tracts in Math (1993). MR**94c:35178****[BCT96]**L. Brandolini, L. Colzani, and G. Travaglini,*Average decay of the Fourier transform, and lattice points inside polyhedra*, (preprint) (1996).

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Additional Information

**Alex Iosevich**

Affiliation:
Department of Mathematics, Georgetown University, Washington, DC 20057

Email:
iosevich@math.georgetown.edu

**Kimberly K. J. Kinateder**

Affiliation:
Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435

Email:
kjk@euler.math.wright.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-04879-0

Received by editor(s):
September 17, 1997

Received by editor(s) in revised form:
January 6, 1998

Published electronically:
April 27, 1999

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 1999
American Mathematical Society