Diophantine exponents of lattices

Abstract

In this paper we define Diophantine exponents of lattices and investigate some of their properties. We prove transference inequalities and construct some examples with the help of Schmidt's subspace theorem.

Multiparametric geometry of numbers and its application to splitting transference theorems

“…In this setting the phenomenon of transference can also be observed. The following result was obtained in [7].…”

Linear forms of a given Diophantine type and lattice exponents

“…that Ω d = [0, +∞]. However, until recently the only examples of lattices with positive finite ω(Λ) known to the author were the ones described in [5]. Those lattices give the values ab cd , a, b, c ∈ N,…”

“…When proving Theorem 1 we do not control the dual lattice, so, the only nonzero pairs ω(Λ), ω(Λ * ) currently known to the author are (ω, +∞), where ω is of the form (1). Moreover, the corresponding examples described in [5] have a certain flaw, as in each of them the dual lattice has some nonzero points in the coordinate planes, so that the condition ω(Λ * ) = +∞ is provided by a kind of degeneracy. It would be more interesting to construct lattices that are totally irrational, i.e.…”

“…For instance, this holds for any lattice of a complete module in a totally real algebraic extension of Q (see [2]). Due to Schmidt's subspace theorem [3] we also have ω(Λ) = 0 for a much wider class of algebraic lattices satisfying certain independence conditions (see [4], [5]). It is worth mentioning that, same as with real numbers, such an algebraic lattice behaves as an average unimodular lattice.…”

Multiplicative parametric geometry of numbers and transference theorems for lattice exponents

“…Recently in [6] a transference theorem for Diophantine exponents of lattices was proved. Let L d denote the space of unimodular lattices in…”