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On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation

Michael Fuchs, Dong Han Kim (2016)

Acta Arithmetica

We give a necessary and sufficient condition such that, for almost all s ∈ ℝ, ||nθ - s|| < ψ(n) for infinitely many n ∈ ℕ, where θ is fixed and ψ(n) is a positive, non-increasing sequence. This can be seen as a dual result to classical theorems of Khintchine and Szüsz which dealt with the situation where s is fixed and θ is random. Moreover, our result contains several earlier ones as special cases: two old theorems of Kurzweil, a theorem of Tseng and a recent...

On metric theory of Diophantine approximation for complex numbers

Zhengyu Chen (2015)

Acta Arithmetica

In 1941, R. J. Duffin and A. C. Schaeffer conjectured that for the inequality |α - m/n| < ψ(n)/n with g.c.d.(m,n) = 1, there are infinitely many solutions in positive integers m and n for almost all α ∈ ℝ if and only if ∑ n = 2 ∞ ϕ ( n ) ψ ( n ) / n = ∞ . As one of partial results, in 1978, J. D. Vaaler proved this conjecture under the additional condition ψ ( n ) = ( n - 1 ) . In this paper, we discuss the metric theory of Diophantine approximation over the imaginary quadratic field ℚ(√d) with a square-free integer d < 0, and show that a Vaaler...

Simultaneous inhomogeneous Diophantine approximation of the values of integral polynomials with respect to Archimedean and non-Archimedean valuations

Ella I. Kovalevskaya, Vasily Bernik (2006)

Acta Mathematica Universitatis Ostraviensis

We prove an analogue of the convergence part of Khintchine’s theorem for the simultaneous inhomogeneous Diophantine approximation on the Veronese curve ( x , x 2 , ... , x n ) with respect to the different valuations. It is an extension of the author’s earlier results.

Some algebraic and homological properties of Lipschitz algebras and their second duals

F. Abtahi, E. Byabani, A. Rejali (2019)

Archivum Mathematicum

Let ( X , d ) be a metric space and α > 0 . We study homological properties and different types of amenability of Lipschitz algebras Lip α X and their second duals. Precisely, we first provide some basic properties of Lipschitz algebras, which are important for metric geometry to know how metric properties are reflected in simple properties of Lipschitz functions. Then we show that all of these properties are equivalent to either uniform discreteness or finiteness of X . Finally, some results concerning the character...

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