Sidon sets

Interview ---------

What field is your thesis in?

My thesis, entitled "Sets with small sumsets and Sidon sets: a study of two extremes", lies within number theory, and more specifically additive combinatorics. I completed it under the supervision of Alain Plagne (École polytechnique) and Anne de Roton (Université de Lorraine). I am currently a temporary teaching and research fellow at the École normale supérieure Paris-Saclay.

Why did you choose this topic?

I chose this topic because it is very concrete, yet calls for a great deal of imagination. Problems in additive combinatorics are very simple to state and require no particularly advanced theory merely to formulate. Yet they often prove remarkably resistant, and some have seen no progress for several decades. New and ingenious ideas are therefore needed, sometimes drawing on highly abstract theories with no apparent direct connection to the original question. That is what appeals to me.

How does this thesis fit into your plans?

First of all, my three years as a doctoral student were wonderful, so the PhD itself was a great experience. Furthermore, I intend to become a university lecturer and researcher, which is not possible without a doctorate. Finally, defending my thesis notably enabled me to work as a temporary teaching and research fellow for the past three years, and therefore to continue my research as well.

In mathematics, a Sidon set is a set in which all the differences between two elements are distinct. This means that one cannot find four distinct elements a, b, c and d such that a – c = d – b, or equivalently a + b = c + d.

Equivalently, it is a set in which all sums of two elements are distinct. These are extremely natural sets, and the first mathematician to study them in the literature was the Hungarian Simon Sidon (1892‒1941), in 1932; they are named after him. At the time, he was working on the theory of Fourier series and raised several questions about Sidon sets. The most famous was this: what is the maximum size of a Sidon set among the first n natural numbers?

Asymptotic results ----------------------------

What is the maximum number of integers below 100 that can be chosen so that they form a Sidon set? This is the kind of question that arises in this field.