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October 2, 20250 citationsOpen Access

On integer points inside a randomly shifted polyhedron

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ATAleksandr Tokmachev

Key Points

  • The expected number of lattice points inside a randomly translated polyhedron equals its volume.
  • This work investigates variance and higher moments of the lattice point distribution in polytopes.
  • Concentration of measure phenomena are discussed, highlighting how random shifts affect lattice point counts.
  • The study provides foundational insights for understanding spatial distributions in higher dimensions.

Abstract

Consider a convex body C Rᵈ. Let X be a random point with uniform distribution in 0, 1ᵈ. Define XC as the number of lattice points in Zᵈ inside the translated body C + X. It is well known that E XC = vol (C). A natural question arises: What can be said about the distribution of XC in general? In this work, we study this question when C is a polyhedron with vertices at integer points.

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Cite This Study

Aleksandr Tokmachev (2025) studied this question.

synapsesocial.com/papers/68de5da283cbc991d0a2069dhttps://doi.org/10.48550/arxiv.2507.09355
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