smalldivisor

A small divisor refers to a quantity that appears in the denominators of perturbation expansions and can become arbitrarily close to zero, causing convergence or stability problems in analysis. The term is most often used in dynamical systems, celestial mechanics, and partial differential equations.

In many problems, solutions are built by expanding in Fourier modes or by solving linearized equations with terms that involve denominators of the form ω·k or differences of nearly equal eigenvalues. If these denominators are small for some integer vector k or mode, the corresponding terms can become large and spoil convergence of the series or the validity of the perturbative method. This phenomenon is known as the small divisor problem.

A central way to control small divisors is to impose arithmetic non-resonance conditions on frequency vectors. The most common is the Diophantine condition: there exist constants γ > 0 and τ ≥ n−1 such that |k·ω| ≥ γ|k|^{-τ} for all nonzero integer vectors k. In one dimension, Brjuno conditions provide refinements. Imposing such conditions helps ensure convergence of perturbation schemes and the persistence of quasi-periodic structures.

The small divisor problem is a key obstacle in Kolmogorov-Arnold-Moser (KAM) theory, which studies the persistence of invariant tori under small perturbations. It also appears in the analysis of partial differential equations and in normal form theory, where near-resonant interactions must be controlled.

See also: resonance, Diophantine approximation, KAM theory, cohomological equation.