Paul Le Meur

]]>The bound you mention for K(mn) with K a general trace function is due to Ping Xi in his thesis using exactly the method you describe and using Katz’s theory of multiplicative convolution to handle the algebraic geometry.

Inspired by this, I worked out a generalization to K(f(m,n)) for an arbitrary polynomial f, and even to arbitrary sheaves and higher dimensions. It should essentially replace 1/4 with 1/2-epsilon in your algebraic regularity lemma while also generalizing from definable functions to trace functions (and precisely 1/2 in some cases). I have not finished writing it up, though.

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