On connectedness in the parametric geometry of numbers

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.

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References in corpus (2)

• On Schmidt and Summerer parametric geometry of numbers
• Spectrum of the exponents of best rational approximation