“…Hlawka [8] obtained a general dimensional asymptotic formula, which in three dimensions yields an error of size (1); see also Krätzel [13]. Under the C ∞ hypothesis of a convergent Taylor series, the error term in the asymptotic formula has been improved, most recently by Müller [18].…”
mentioning
confidence: 99%
“…In Müller [18] the differential inequality assumed is that the Gaussian curvature does not vanish. We can regard (3) as a corresponding quantitative bound.…”
“…Hlawka [8] obtained a general dimensional asymptotic formula, which in three dimensions yields an error of size (1); see also Krätzel [13]. Under the C ∞ hypothesis of a convergent Taylor series, the error term in the asymptotic formula has been improved, most recently by Müller [18].…”
mentioning
confidence: 99%
“…In Müller [18] the differential inequality assumed is that the Gaussian curvature does not vanish. We can regard (3) as a corresponding quantitative bound.…”
“…These are due to W. Müller [14] (who improved earlier results by E. Hlawka [5] and Krätzel and Nowak [10], [11]), and the second named author [15], respectively.…”
Abstract.For a convex body B in R 3 which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy P B (t) (number of integer points minus volume) of a linearly dilated copy √ tB is estimated from below. On the basis of a recent method of K. Soundararajan [16] an Ω -bound is obtained that improves upon all earlier results of this kind.
“…An alternative proof of this result can be found as Satz 5.15 in E. Krätzel's book [11]. By some very hard analysis, W. Müller [16] managed to rene this estimate up to P B (t) = O t 63/43+ε .…”
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.
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