Positive semigroups in lattices and totally real number fields

TL;DR

This paper studies the structure and distribution of gaps in positive semigroups within lattices and applies these findings to bounds on the Weil height of totally positive sub-semigroups in ideals of totally real number fields.

Contribution

It introduces bounds on the successive minima of positive semigroups in lattices and extends these results to number fields, connecting lattice theory with algebraic number theory.

Findings

Bounds for the distribution of gaps in lattice semigroups.

Analogous bounds for Weil height in number fields.

Generalizations of Minkowski's successive minima theorem.

Abstract

Let $L$ be a full-rank lattice in $\mathbb R^d$ and write $L^+$ for the semigroup of all vectors with nonnegative coordinates in $L$. We call a basis $X$ for $L$ positive if it is contained in $L^+$. There are infinitely many such bases, and each of them spans a conical semigroup $S(X)$ consisting of all nonnegative integer linear combinations of the vectors of $X$. Such $S(X)$ is a sub-semigroup of $L^+$, and we investigate the distribution of the gaps of $S(X)$ in $L^+$, i.e. the points in $L^+ \setminus S(X)$. We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of $L^+$ and of $L^+ \setminus S(X)$, for which we produce bounds in the spirit of Minkowski's successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil height of totally positive sub-semigroups of ideals in totally real number fields.