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.

]]>I have published a book on Henri Kowalski (title : “Dans le sillage de Chopin : le pianiste Henri Kowaski”)in 2014, 200 pages and a lot of pictures, from their (Henri and Gabriel) arrival in Dinan and all Henri Kowalski’s career, even, of course in Australia. This book has been published by Le Pays de Dinan et la Bibliothèque de Dinan where you can find Kowalski’s archives, mail : bm@dinan.fr. You’ll find more informations if you type Henri Kowalski et Marie-Claire Mussat.Of course you can order it.

We are preparing a CD with one of the greatest pianist of to-day : François Dumont who gave a recital(Kowalski, Tellefsen, Chopin) at the Paris Polish Library after my lecture in Frebruary and will play again in Dinan in June.

Best regards,

Prof. Marie-Claire Mussat

Musicologist

Emeritus Professor at the Université of Rennes