Publisher: Cambridge University Press   (Total: 387 journals)

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 Forum of Mathematics, PiNumber of Followers: 1     Open Access journal ISSN (Online) 2050-5086 Published by Cambridge University Press  [387 journals]
• CHARACTER LEVELS AND CHARACTER BOUNDS

• Authors: ROBERT M. GURALNICK; MICHAEL LARSEN, PHAM HUU TIEP
Abstract: We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.
PubDate: 2020-01-24T00:00:00.000Z
DOI: 10.1017/fmp.2019.9
Issue No: Vol. 8 (2020)

• MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN
SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS

Abstract: We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ , where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$ . In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$ .In particular, we find that $\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$ . This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of $\sum _{n\leqslant x}f(n)$ .The proofs develop a connection between $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ and the $q$ th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.
PubDate: 2020-01-20T00:00:00.000Z
DOI: 10.1017/fmp.2019.7
Issue No: Vol. 8 (2020)

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