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Preprints still in development

Halasz in short intervals
(with Adam Harper, Kaisa Matomaki, and Maksym Radziwill)

We show Halasz's Theorem in intervals of length \( x^{7/12} \) (removing integers with a prime factor larger than the length of the interval). The method extends to intervals of length \( >x^{1/2+\epsilon} \) but restricting to \( y\)-smooth integers where \( y\) is significantly smaller than the length of the interval.

In this article we study the similarities in the anatomies of integers and permutations.