On commuting circle mappings and simultaneous Diophantine approximations

Dedicated in friendship to A. Haefliger     

Sequences of Diophantine approximations

We sharpen a technique of Gelfond to show that, in a sense, the only possible gap-free sequences of “good” Diophantine approximations to a fixed α ∈ C are trivial ones. For example, suppose that a > 1 and that (δ_n)_n=1^∞ and (σ_n)_n=1^∞ are two positive, strictly increasing unbounded sequences satisfying δ_n+1 ≤ aδ_n and σ_n+1 ≤ aσ_n. If there is a sequence of nonzero polynomials P_n ∈ Z[x] with deg P_n ≤ δ_n, deg P_n + log height P_n ≤ σ_n, and ∣P_n(α)∣ ≤ e^{?(2a+1)δ_nσ_n}, then each P_n(α) = 0.     

Diophantine approximations and foliations

In this paper we indicate the proof of an effective version of the Green-Griffiths conjecture for surfaces of general type and positive second Segre class (i.e.c ₁ ² >c ₂). Naturally this effective version is stronger than the Green-Griffiths conjecture itself.     

Badly Approximable Matrices and Diophantine Exponents

Doklady Mathematics - This paper is a survey of results concerning different kinds of Diophantine exponents. Special attention is paid to the transference principle and the generalization of the...     

Best simultaneous Lp approximations

In this paper we study simultaneous approximation of n real-valued functions in L _p[a, b] and give a generalization of some related results.     

Diophantine approximations of log 2 and other logarithms

We describe a new approach to estimating μ(log 2), without improving Rukhadze’s result (1987). We find estimates for approximations to the number log 2 by numbers from the field ?(√2), to the number log((√5?1)/2) by numbers from the field ?(√5), and to some other numbers.     

Location of best simultaneous approximations

Amir and Ziegler have proved that a real normed space E of dimension ≥ 3 is an inner product space if and only if, for any x,y ε E and any two dimensional subspace M of E, the expression max(‖x — w‖,‖y — w‖) attains its minimum in some point w of each segment [u,v], with u and v, respectively, best approximations to x and y from M. We extend this result to expressions π(‖x — w‖,‖y — w‖), where π denotes a monotonic norm in R²     

Unconventional height functions in simultaneous Diophantine approximation

Simultaneous Diophantine approximation is concerned with the approximation of a point \(\mathbf x\in \mathbb R^d\) by points \(\mathbf r\in \mathbb Q^d\), with a view towards jointly minimizing the quantities \(\Vert \mathbf x - \mathbf r\Vert \) and \(H(\mathbf r)\). Here \(H(\mathbf r)\) is the so-called “standard height” of the rational point \(\mathbf r\). In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.     

On Diophantine approximations of Ramanujan type q-continued fractions

We shall consider arithmetical properties of the q-continued fractions

     

Additive functions and simultaneous rational approximations

V. Kapsukas Vilnius State University. Translated from Litovskii Matematichskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 1, pp. 129–147, January–March, 1989.     

Strong approximations of three-dimensional Wiener sausages

We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L ²-norm estimates between the Wiener sausage and the Brownian intersection local times. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886, T 043037 and K 61052.     

Products of simultaneous approximations of rational numbers

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