It was shown by Bourgain and Rudnick in 2010 that for a curve of non-vanishing curvature on the standard flat 2-torus there are no eigenfunctions of arbitrarily large eigenvalues which vanish on it. We show that for any small circle on the 2-sphere there are no zonal spherical harmonics of arbitrarily large eigenvalues which vanish on it. Equivalently, only a finite number of Legendre polynomials can share a non-zero root. The conjecture by Stieltjes is that Legendre polynomials have no common non-zero roots. The talk is based on joint works with Adi Weller Weiser and Borys Kadets.
