Web Page of the Ottawa-Carleton Number Theory Seminar

Year 2018-2019

Abstract: A singular point on a plane conic defined over Q is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are abscissa of such points on the parabola y=x^2. In this talk, we provide a measure of transcendence for singular points on conics defined over Q which, in these two cases, improves on the measure obtained by Adamczewski et Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.

Title: On Waring's problem in short intervals: the case of four fourth powers

Abstract: Consider the set S of natural numbers that can be written as a sum of four 4th powers. How large is the spacing between two elements of this set? It can be arbitrarily large, as a consequence recent work by the author. In this talk we show, on the other hand, how do the Circle Method of G.H.Hardy and J.E.Littlewood and a strategy of S.Daniel provide nontrivial upper bounds on these gaps. This work is motivated by an arithmetic study of the values of a biquadratic theta function.