Stability versions of uncertainty principles are established on compact Riemannian manifolds for singular potentials by replacing homogeneity with a quantitative spectral condition, producing sharp bounds that quantify deterioration from the classical case.
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The Möbius function restricted to squarefrees up to R has Fourier ratio at least R to the power -1/4 minus epsilon, forcing any uniform learner for the class to use Omega(R) samples.
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Uncertainty principles and singular potentials
Stability versions of uncertainty principles are established on compact Riemannian manifolds for singular potentials by replacing homogeneity with a quantitative spectral condition, producing sharp bounds that quantify deterioration from the classical case.
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Arithmetic functions and learning theory
The Möbius function restricted to squarefrees up to R has Fourier ratio at least R to the power -1/4 minus epsilon, forcing any uniform learner for the class to use Omega(R) samples.