We consider inhomogeneous Diophantine approximation for formal Laurent series over a finite base field. We establish an analogue of a strong law of large numbers due to W. M. Schmidt with a better error term than in the real case. A special case of our result improves upon a recent result by H. Nakada and R. Natsui and completes a result of M. M. Dodson, S. Kristensen, and J. Levesley. Moreover...

We show that whenever δ > 0 and constants λi satisfy some necessary conditions, there are infinitely many prime triples p1, p2, p3 satisfying the inequality |λ0 + λ1p1 + λ2p2 + λ3p3| < (max pj)−2/9+δ. The proof uses Davenport–Heilbronn adaption of the circle method together with a vector sieve method. 2000 Mathematics Subject Classification. 11D75, 11N36, 11P32.

— We show that Y. Cheung’s general Z-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an ...

An inhomogeneous version of a general form of the Jarn k-Besicovitch Theorem is proved. Dedicated to Professor F. Chong for his 80th birthday 1. Introduction In some respects, inhomogeneous Diophantine approximation is rather diierent from homogeneous Diophantine approximation. Results in the former, where the additional variables ooer extràdegrees of freedom', are sometimes sharper or easier t...

The theory of inhomogeneous Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the Groshev type theory is established for all such manifolds. Our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non-degenerate manifolds established to date. The results have natur...

Using a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — arf < c, r(q — fir) < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals a and /3, provided that c > 1/2.6394. This improves an earlier result of Davenport, who shows that the same conclusion holds if c > 1/46"" = 1/2.6043 • • •.

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions: (i) Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem? (ii) Is it useful to combine this approach with traditional approaches to Diophantine e...

We prove that if (x, y, n, q) 6= (18, 7, 3, 3) is a solution of the Diophantine equation (xn−1)/(x−1) = y with q prime, then there exists a prime number p such that p divides x and q divides p − 1. This allows us to solve completely this Diophantine equation for infinitely many values of x. The proofs require several different methods in diophantine approximation together with some heavy comput...

We give a new proof of the validity of Cornacchia’s algorithm for finding the primitive solutions (u, v) of the diophantine equation u + dv = m, where d and m are two coprime integers. This proof relies on diophantine approximation and an algorithmic solution of Thue’s problem.