Distribution of the error term for the number of lattice points inside a shifted ball

Abstract

Define

$$\begin{array}{*{20}c} {N_x (R)\, = \,\# \,\left\{ {n\, \in \,Z^d \,|\,|x\, - \,n|\, \leqslant \,R} \right\}} & {(x\, \in \,[0,\,1[^d )} \\ \end{array}$$

((1.1))

and

$$F_x (R)\, = \,R^{ - \frac{{d\, - \,1}} {2}} \,[N_x (R)\, - \,R^d \,\text{Vol}_d \,B(1)]$$

((1.2))

where B(1) is the unit ball in R ^d. Thus N _x (R) is the number of lattice points in Z ^d in the R-ball centered at x and F _x (R) is the normalized difference with the volume of the ball. We are interested in the problem of the distribution of F _x (R) as a function of R for given x.