I submitted the paper in 
<a href="https://arxiv.org/abs/math/0609045">this form</a> in 2006 to Publications IHES; it was reviewed by Artur Avila, and 
<a href="Comments on Bounds I.pdf">here</a> is the report that I received. (The report makes it clear that the I knew its author). Artur had a clear and thorough understanding of the context and significance of the paper, and, as he says, we had a long conversation about the mathematics, and then he was able to understand how it worked. He made some suggestions about how to improve the exposition that I basically agreed with, and then I was busy with teaching and other work and I put it off...and put it off...and put it off...and put it off. 
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Finally, in 2022, I set myself to improving the exposition along the lines that Artur had suggested, and in accordance with my own vision of how the paper should be. The 
<a href="BoundsI-working-draft.pdf">new version</a> will soon be resubmitted for publication.