The lattice point discrepancy of a torus in ℝ3

Abstract

This article provides an asymptotic formula for the number of integer points in the three-dimensional body   x y z   = t   (a + r cos α) cos β (a + r cos α) sin β r sin α   , 0 α, β < 2π, 0 r b,for xed a > b > 0 and large t.

Lattice points in the 3-dimensional torus

“…A particularly symmetric example, considered in [Now08a], is the solid torus (1.1) T = (x, y, z) ∈ R 3 : ρ ′ − x 2 + y 2 2 + z 2 ≤ ρ 2 where 0 < ρ < ρ ′ are fixed constants. Its volume is 2π 2 ρ 2 ρ ′ but the natural main term to approximate the number of lattice points in the R-scaled torus…”

“…Nowak [Now08a] who obtained E(R) = O R 11/8+ǫ . He wrote M(R) as a series depending on elementary functions but it is equivalent to our statement after substituting the asymptotic formula for J 1 .…”

“…In §3 an application of the stationary phase principle allows to interpret the formula for E(R) as an exponential sum. In [Now08a] this aim is reached indirectly thanks to the truncated Hardy-Voronoï formula for the circle problem [Ivi03] (the circles appear because each horizontal section of T is a corona). The advantages of our approach are that it is applicable to other problems, it is self-contained and it reveals the main term in a more transparent way.…”

A mean-square bound concerning the lattice discrepancy of a torus in ℝ3

“…For the remainder, it has been shown by D. Popov [13] that Δ T (t) t 3 2 − 1 286 +ε , and by the second named author [12] that Δ T (t) t 3 2 − 1 8 +ε . It is the objective of this paper to show that again Δ T (t) t 1+ε in meansquare.…”

“…

For positive constants a > b > 0, let PT (t) denote the lattice point discrepancy of the body tT a,b , where t is a large real parameter and T = T a,b is bounded by the surfaceIn a previous paper [12] it has been proved thatwhere F a,b (t) is an explicit continuous periodic function, and the remainder satisfies the ("pointwise") estimate Δ T (t) t 11/8+ε . Here it will be shown that this error term is only t 1+ε in mean-square, i.e., that

for any ε > 0.

…”