Abstract
For d-dimensional ellipsoids E with d≥5 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order \(\mathcal{O}(r^{d-2})\) for general ellipsoids and up to an error of order o(r d-2) for irrational ones. The estimate refines earlier bounds of the same order for dimensions d≥9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],m∈ℤd of positive definite irrational quadratic forms Q of dimension d≥5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms.
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Mathematics Subject Classification (1991)
11P21
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Götze, F. Lattice point problems and values of quadratic forms. Invent. math. 157, 195–226 (2004). https://doi.org/10.1007/s00222-004-0366-3
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DOI: https://doi.org/10.1007/s00222-004-0366-3