For an irrational real number$\alpha$ and real number $\gamma$ we consider the inhomogeneous approximation constant$$M(\alpha ,\gamma ):=\underset{\left|n\right|\to \infty }{lim\; inf}\left|n\right|\left|\right|n\alpha -\gamma \left|\right|$$via the semi-regular negative continued fraction expansion of $\alpha$$$\alpha =\frac{1}{{}_{}}$$

In this paper we describe the set of conjugacy classes in the group $\mathrm{SL}(n,\mathbb{Z})$. We expand geometric Gauss Reduction Theory that solves the problem for $\mathrm{SL}(2,\mathbb{Z})$ to the multidimensional case, where $\varsigma$-reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in $\mathrm{GL}(n,\mathbb{Z})$ in terms of multidimensional Klein-Voronoi continued fractions.

Given a binary recurrence ${u_n}_{n\ge 0}$, we consider the Diophantine equation $u_{n_1}^{x_1}\cdots u_{n_L}^{x_L}=1$ with nonnegative integer unknowns $n_1,...,n_L$, where $n_i\ne n_j$ for 1 ≤ i < j ≤ L, $max|x_i|:1\le i\le L\le K$, and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete 'multiplicative' zero-one law is established akin to the 'simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile 'cross fibering principle'. In a nutshell it enables us to 'lift' zero-one laws to higher...

We consider the Tribonacci sequence $T:={T_n}_{n\ge 0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n+3}=T_{n+2}+T_{n+1}+T_n$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan’s classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd positive integers (Nesterenko, 2011).