Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 102, 9 pages      arXiv:2508.17551      https://doi.org/10.3842/SIGMA.2026.102

Integral Points and Volume of Integral-Integral Affine Manifolds

Oded Elisha a, Yael Karshon a and Yiannis Loizides b
a) School of Mathematical Sciences, Tel Aviv University, Israel
b) Department of Mathematical Sciences, George Mason University, USA

Received February 17, 2026, in final form October 11, 2026; Published online October 29, 2026

Abstract
Using rational Ehrhart theory and Fourier analysis, we give an elementary proof that the volume of a closed manifold equipped with an integral-integral affine structure equals the number of its integral points.

Key words: integral affine structure; Ehrhart theory.

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