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On connectedness in the parametric geometry of numbers

Yuming Wei and Han Zhang

Vol. 13 (2024), No. 4, 299–315
DOI: 10.2140/cnt.2024.13.299
Abstract

Via multilinear algebra, we formulate a criterion for connectedness in the parametric geometry of numbers in terms of pencils, which are certain algebraic varieties in the space of matrices. As a consequence, we obtain a connectedness result for generic lattices arising from Diophantine approximation on analytic submanifolds, and sharpen Schmidt and Summerer’s results of connectedness on simultaneous Diophantine approximation and approximation by linear forms.

Keywords
geometry of numbers, connectedness, pencils
Mathematical Subject Classification
Primary: 11H06, 11J13, 37B05
Milestones
Received: 23 April 2024
Revised: 19 August 2024
Accepted: 18 September 2024
Published: 27 September 2024
Authors
Yuming Wei
Yau Mathematical Sciences Center
Tsinghua University
Beijing
China
Han Zhang
School of Mathematical Science
Soochow University
Suzhou
China