A uniform nilsequence Wiener-Wintner theorem for bilinear ergodic averages

Abstract

We show that a $k$-linear pointwise ergodic theorem on an ergodic measure-preserving system implies a uniform $k$-linear nilsequence Wiener-Wintner theorem on that system. The assumption is known to hold for arbitrary systems and $k=2$ (due to Bourgain) and for distal systems and arbitrary $k$ (due to Huang, Shao, and Ye).

v2: 4 pages, characterization of good weights for L^2 convergence added, uniformity seminorm in the main result corrected

References in corpus (2)

• Pointwise double recurrence and nilsequences
• Uniformity seminorms on ${\ell }^{\infty }$ and applications

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• Pointwise double recurrence and nilsequences
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• On the polynomials homogeneous ergodic bilinear averages with Liouville and Möbius weights
• A good universal weight for multiple recurrence averages with commuting transformations in norm