On Diophantine inequalities

Abstract

In this master thesis, we study Diophantine inequalities. Our aim is to solve inequalities involving polynomials with arbitrary real coefficients, where the ratio of two coefficients is irrational. In 1929, it was conjectured by Oppenheim that the inequality |\alpha_1x_1^2 + \ldots + \alpha_nx_n^2|<\delta, provided n \geq 5, \alpha_1,\ldots,\alpha_n\in\R, not all of the same sign and such that two coefficients have irrational ratio, is soluble for all \delta>0. A very interesting topic is to make this result quantitative. We study a quantitative result of Bourgain on quadratic ternary diagonal forms for one parameter families and a generalisation to generic ternary diagonal forms by Schindler. We generalise this previous work and study general Diophantine inequalities |G_k(x_1,x_2) - \alpha_3x_3^l|< \delta, with G_k a binary form of degree k\geq 3 and coefficients in \R, l\neq k and \alpha_3\in\R. The goal is to find non-trivial solutions (x_1,x_2,x_3)\in\Z where x_1,x_2 are of size N^l and x_3 of size N^k. We obtain results for the cases G_k(x_1,x_2) = x_1^k - \alpha_2x_2^k and G_k(x_1,x_2) = x_1^k + \alpha_1x_1^{k/2}x_2^{k/2}+ \alpha_2x_2^k and formulate a conjecture about a general polynomial G_k.