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Distribution of integer lattice points in a ball centred at a Diophantine point. (English) Zbl 1252.11075
Summary: We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point. Starting from Poisson summation, careful choices of parameters are required for smoothing process, and the crucial ingredient is the mean square value of exponential sums. The goal in this problem is to determine whether the distribution agrees with Berry-Tabor conjecture.

MSC:
11P21 Lattice points in specified regions
42B05 Fourier series and coefficients in several variables
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