On Some Diophantine Inequalities Involving the Exponential Function

Abstract

It is well known that for any real number θ there are infinitely many positive integers n such thatHere ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

The Distribution of Lattice Points in Elliptic Annuli

“…Of particular interest to us are the following regimes: (1) The microscopic regime ρt is constant. It was conjectured by Berry and Tabor [7] that the statistics of N B (t, ρ) are Poissonian.…”

On Baker type lower bounds for linear forms

“…The underlying idea behind our treatment is well known already from Baker's work [1]. Namely, the idea, see formula (22) in [1], is to fix the parameter n j with the corresponding individual height H j (in our notation).…”

“…(Note that the exponential function belongs to the class of Siegel's Efunctions.) By applying Theorem 3.4 of the present paper the authors in [4] proved substantial improvements of the explicit versions, see Mahler [9] and Sankilampi [10], of Baker's work [1] about exponential values at rational points. In particular, the dependence on m is improved.…”

“…Throughout this work, let I denote an imaginary quadratic field with Z I it's ring of integers. By an explicit Baker type lower bound we mean any positive lower bound With the assumption that γ 0 , γ 1 , ..., γ m ∈ Q * are distinct, Baker [1] proved that there exist positive constants δ 1 , δ 2 and δ 3 such that Here we note that the constants δ 1 , δ 2 , δ 3 in Baker's work [1] are not explicitly given. Mahler [9] made Baker's result completely explicit.…”

“…Namely, the idea, see formula (22) in [1], is to fix the parameter n j with the corresponding individual height H j (in our notation). In our work we shall express this phenomenon first in a nutshell, see (4.10), and then in a refined form, see (4.14).…”