Counter-examples in parametric geometry of numbers

Martin Rivard-Cooke, Damien Roy
2020 Acta Arithmetica  
1. Introduction. The basic object of Diophantine approximation is rational approximation to points u in R n . This is generally measured by elements of the extended real line [−∞, ∞] called exponents of approximation to u. The spectrum of a family of exponents (µ 1 , . . . , µ m ) is the subset of [−∞, ∞] m consisting of all m-tuples (µ 1 (u), . . . , µ m (u)) as u varies among the points of R n with linearly independent coordinates over Q. In all cases where such a spectrum has been explicitly
more » ... has been explicitly determined, its trace on R m (the set of its finite points) can be expressed as the set of common solutions of a finite system of polynomial inequalities (called transference inequalities). In particular, this trace is a semialgebraic subset of R m , namely a finite union of such solution sets. It is natural to ask if this is always so. A general study of spectra is proposed in [7] . It is based on parametric geometry of numbers and the observation, due to Schmidt and Summerer [8], that the standard exponents of approximation to a point u ∈ R n can be computed from the knowledge of the successive minima of a certain oneparameter family of convex bodies in R n . Using the equivalent formalism of [5], we choose the family
doi:10.4064/aa191217-9-4 fatcat:mpwrqcgcgbhnhatna63s23ov2a