There is no analogue to Jarník’s relation for twisted Diophantine approximation

Abstract

Jarník gave a relation between the two most classical uniform exponents of Diophantine approximation in dimension 2. In this paper we consider a twisted case, between the classical and the multiplicative one, and we show that no analogue to Jarník's relation holds.

Transference Theorems for Diophantine Approximation With Weights

“…The case m = 1 is slightly more difficult. It appears that in this case ω(Θ) cannot be greater than ρ −1 n (unless Θ is rational), which eliminates the first alternative in (7). In fact, a stronger statement holds.…”

“…Thus, for n = 1 we have ρ n = 1 and ω(Θ) 1 > σ m = σ m /ρ n , i.e. the second alternative in (7) is inconsistent.…”

“…As for the uniform exponents, it appears that both for m = 1 and n = 1 exactly one of the inequalities (7) survives. For n = 1 this fact easily follows from Minkowski's convex body theorem, which gives for every m and n the trivial bound ω(Θ) ω(Θ) 1.…”

“…Note that due to the trivial inequalities ω η η η ω η η η , ω ω both (15) and (16) provide lower estimates for ω η η η in terms of ω. One can easily check that the one provided by (15) is weaker than the one provided by (16). However, there is a small disadvantage in the latter caused by the condition onω.…”

Теория Чисел И Приложения, 1

“…[10]- [12]), естественно возникает вопрос описания соответствующего спектра -подмножества (R∪{∞}) 2 , заметаемого парами ( (Λ), (Λ * )), если Λ пробегает все пространство решеток ранга , содержащихся в R .…”