Diophantine approximation on rational quadrics
نویسنده
چکیده
We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.
منابع مشابه
On the Diophantine Equation x^6+ky^3=z^6+kw^3
Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is n...
متن کاملDiophantine approximation and Diophantine equations
The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
متن کاملon ”NUMBER THEORY AND MATHEMATICAL PHYSICS” On recent Diophantine results
Diophantus of Alexandria was a greek mathematician, around 200 AD, who studied mathematical problems, mostly geometrical ones, which he reduced to equations in rational integers or rational numbers. He was interested in producing at least one solution. Such equations are now called Diophantine equations. An example is y − x = 1, a solution of which is (x = 2, y = 3). More generally, a Diophanti...
متن کاملRational Points on Pencils of Conics and Quadrics with Many Degenerate Fibres
For any pencil of conics or higher-dimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics over Q, which is a consequence of recent advances in additive combinatorics.
متن کاملDiophantine Approximation and Algebraic Curves
The first topic of the workshop, Diophantine approximation, has at its core the study of rational numbers which closely approximate a given real number. This topic has an ancient history, going back at least to the first rational approximations for π. The adjective Diophantine comes from the third century Hellenistic mathematician Diophantus, who wrote an influential text solving various equati...
متن کامل