Mixing automorphisms of measure-preserving systems are among the best-understood objects in ergodic theory. The spectral properties of the automorphism itself — its eigenvalues and eigenfunctions — characterize its mixing behavior. The symmetric tensor square of a mixing automorphism is a derived object: it acts on symmetric functions of pairs of points and inherits spectral properties from the original system.
Kolmogorov and Rokhlin asked: can the symmetric tensor square have simple spectrum — every eigenvalue appearing with multiplicity one? The question has remained open because simplicity of the tensor square is far more restrictive than simplicity of the original automorphism. The tensor square naturally produces multiplicity from the pairing structure, and eliminating all of it requires precise spectral control.
The paper constructs explicit examples. Using near-Sidon constructions — sets of integers whose pairwise sums are nearly all distinct — the authors build mixing automorphisms whose symmetric squares have simple spectra. The Sidon-like property ensures that the spectral contributions from different pairs don't overlap, preventing the multiplicity that the tensor structure would generically produce.
The result is an existence proof with an explicit construction, answering a question from the 1960s. The structural content: simplicity of the tensor square is possible but requires the original automorphism's spectrum to satisfy an arithmetic condition (near-Sidon). The dynamical property (simple spectrum of the square) is controlled by a number-theoretic property (distinctness of pairwise sums) of the spectral data. Ergodic theory meets additive combinatorics.