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  • Bernardo González Merino (a1) and Matthias Henze (a2)

One of the most fruitful results from Minkowski’s geometric viewpoint on number theory is his so-called first fundamental theorem. It provides an optimal upper bound for the volume of a $0$ -symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of a $0$ -symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.

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1. Aliev, I., Bassett, R., De Loera, J. A. and Louveaux, Q., A quantitative Doignon–Bell–Scarf theorem. Combinatorica (to appear), arXiv:1405.2480.
2. Averkov, G., On maximal S-free sets and the Helly number for the family of S-convex sets. SIAM J. Discrete Math. 27(3) 2013, 16101624.
3. Averkov, G., Krümpelmann, J. and Nill, B., Largest integral simplices with one interior integral point: solution of Hensley’s conjecture and related results. Adv. Math. 274 2015, 118166.
4. Averkov, G. and Weismantel, R., Transversal numbers over subsets of linear spaces. Adv. Geom. 12(1) 2012, 1928.
5. Betke, U., Henk, M. and Wills, J. M., Successive-minima-type inequalities. Discrete Comput. Geom. 9(2) 1993, 165175.
6. van der Corput, J. G., Verallgemeinerung einer Mordellschen Beweismethode in der Geometrie der Zahlen II. Acta Arith. 2 1936, 145146.
7. Doignon, J.-P., Convexity in cristallographical lattices. J. Geom. 3 1973, 7185.
8. Draisma, J., McAllister, T. B. and Nill, B., Lattice-width directions and Minkowski’s 3 d -theorem. SIAM J. Discrete Math. 26(3) 2012, 11041107.
9. Freiman, G. A., Heppes, A. and Uhrin, B., A lower estimation for the cardinality of finite difference sets in R n . In Number Theory, Vol. I (Budapest, 1987) (Colloquium Mathematical Society János Bolyai 51 ), Amsterdam (North-Holland, 1990), 125139.
10. Gruber, P. M., Convex and Discrete Geometry (Grundlehren der Mathematischen Wissenschaften 336 ), Springer (Berlin, 2007).
11. Gruber, P. M. and Lekkerkerker, C. G., Geometry of Numbers, 2nd edn., (North-Holland Mathematical Library 37 ), North-Holland (Amsterdam, 1987).
12. Malikiosis, R., A discrete analogue for Minkowski’s second theorem on successive minima. Adv. Geom. 12(2) 2012, 365380.
13. Minkowski, H., Geometrie der Zahlen (Bibliotheca Mathematica Teubneriana, Teubner (Leipzig–Berlin, 1896), reprinted by Johnson Reprint Corp., New York, 1968.
14. Pikhurko, O., Lattice points in lattice polytopes. Mathematika 48(1–2) 2001, 1524.
15. Rabinowitz, S., A theorem about collinear lattice points. Util. Math. 36 1989, 9395.
16. Ruzsa, I. Z., Additive combinatorics and geometry of numbers. In Proceedings of the International Congress of Mathematicians, Vol. 3, European Mathematical Society (Zürich, 2006), 911930.
17. Scott, P. R., On convex lattice polygons. Bull. Aust. Math. Soc. 15(3) 1976, 395399.
18. Stanchescu, Y. V., On finite difference sets. Acta Math. Hungar. 79(1–2) 1998, 123138.
19. Stanchescu, Y. V., An upper bound for d-dimensional difference sets. Combinatorica 21(4) 2001, 591595.
20. Tao, T. and Vu, V., Additive Combinatorics (Cambridge Studies in Advanced Mathematics 105 ), Cambridge University Press (Cambridge, 2006).
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
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