The t-metric Mahler measures of surds and rational numbers

Abstract

A. Dubickas and C. Smyth introduced the metric Mahler measurewhere M (α) denotes the usual (logarithmic) Mahler measure of α ∈ Q. This definition extends in a natural way to the t-metric Mahler measure by replacing the sum with the usual Lt norm of the vector (M (α1), . . . , M(αN )) for any t 1. For α ∈ Q, we prove that the infimum in Mt(α) may be attained using only rational points, establishing an earlier conjecture of the second author. We show that the natural analogue of this result fails for general α ∈ Q by giving an infinite family of quadratic counterexamples. As part of this construction, we provide an explicit formula to compute Mt(D 1/k ) for a squarefree D ∈ N.

Metric Mahler measures over number fields

“…We say that a number field K is balanced if for every non-zero point x ∈ O K there exists a unit u ∈ O K such that |ux| v ≥ 1 for all Archimedean places v of K. If there exists x ∈ O K for which no such unit exists, then K is called unbalanced. Our main result is a generalization of the proof technique in [9] described above. Theorem 1.1.…”

“…It is clear from the definition that lim t→∞ m K,t (α) = m K,∞ (α). The t-metric Mahler measures over Q have been studied extensively by Dubickas, Smyth, Fili, Jankauskas and the author in an assortment of previous articles [4,5,7,9,[15][16][17][18][19][20]. For example, the author [16] showed that the infimum in the definition of m Q,t (α) is attained for all α ∈ Q and all t > 0.…”

Metric heights on an Abelian group

“…satisfies this property. The work of Jankauskas and the author [6] provides examples, however, in which K α does not replace Q × uniformly on (0, ∞).…”

Optimal factorizations of rational numbers using factorization trees

“…The proof of Theorem 1.2 does provide a method to search a list of candidates for an optimal representation, but it alone gives little information on how to determine a sufficiently small list of candidates. On the other hand, a result of Jankauskas and the first author [5] provides a crucial improvement to Theorem 1.2 when α ∈ Q.…”

“…Optimal representations are important because they encode information about the arithmetic properties of α. For instance, if α is a positive integer with prime factorization given by α = p 1 p 2 · · · p N , the work of [5] asserts that (p 1 , p 2 , . .…”