Geodesic
New bounds in the discrete analogue of Minkowski's second theorem
Discrete analysis (2024)

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arXiv
We adapt an argument of Tao and Vu to show that if $λ_1\le\cdots\leλ_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $Λ\mathbb{R}^d$, and if we set $k=\max\{j:λ_j\le1\}$, then $K$ contains at most $2^k(1+\frac{λ_k}2)^k/λ_1\cdotsλ_k$ lattice points. This provides improved bounds in a conjecture of Betke, Henk and Wills (1993), and verifies that conjecture asymptotically as $λ_k\to0$. We also obtain a similar result without the symmetry assumption.
Publié le : 
Matthew Tointon. New bounds in the discrete analogue of Minkowski's second theorem. Discrete analysis (2024). http://geodesic.mathdoc.fr/item/DAS_2024_a14/
@article{DAS_2024_a14,
     author = {Matthew Tointon},
     title = {New bounds in the discrete analogue of {Minkowski's} second theorem},
     journal = {Discrete analysis},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2024_a14/}
}
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