Integral points and volume of integral-integral affine manifolds
Oded Elisha, Yael Karshon, Yiannis Loizides
TL;DR
This paper proves that for closed manifolds with integral-integral affine structures, the total volume equals the number of integral points, using Ehrhart theory and Fourier analysis.
Contribution
It provides an elementary proof linking volume and integral points for such manifolds, a novel connection in affine geometry.
Findings
Total volume equals the number of integral points for the manifolds considered.
The proof employs rational Ehrhart theory and Fourier analysis techniques.
The approach simplifies previous methods and offers new insights into affine manifolds.
Abstract
We give an elementary proof that, for a closed manifold with an integral-integral affine structure, its total volume and number of integral points coincide. The proof uses rational Ehrhart theory and elementary Fourier analysis to estimate the difference between the total volume and the number of integral points.
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