Javier Cilleruelo, in various papers together with Antonio Córdoba, Jorge Jiménez-Urroz and the speaker, studied how close together lattice points can be on circles and ellipses. These problems really highlight our ignorance about "small solutions" in all sorts of interesting questions. In particular our joint results, even though they make little headway, have turned out to be very difficult to improve despite some serious effort to do so, due to applications in various areas of analysis and mathematical physics.

In recent work, the speaker and Igor Wigman, have studied further distribution questions about lattice points on circles, but now in a range more accessible to standard  methods.

In this talk we will review these problems and theorems, as well as applications, particularly Javier's elegant contributions.