Compactness and contradiction.
Terence Tao

To be published by the AMS

Last updated: Nov 14, 2012

This continues my series of books derived from my blog. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, “An introduction to measure theory“, “Topics in random matrix theory“, and “Higher order Fourier analysis“.

A draft version of the MS can be found here.

Pre-errata (to be corrected in the published version):

  • Page 78: In the first paragraph, “at least the” should be “at least one of the”.  In the second paragraph, “generated by S_k” and “generated by e_1,\ldots,e_{m-1}” should be “generated by S_k, e_m” and “generated by S_k” respectively. In the third paragraph, after the first sentence, add “We may take N to be a normal subgroup of \hbox{ker}(\phi)“.  In the last paragraph, replace “cannot grow polynomially” by “cannot grow exponentially (as otherwise the number of subsums of A^n e_i for i=1,\ldots,d and n=1,\ldots,N would grow exponentially in N, contradicting the polynomial growth hypothesis)”
  • Page 79, footnote 12: replace the first sentence by “Proof: the algebraic integers \alpha^n for natural number n have bounded degree and all Galois conjugates bounded, so the minimal polynomials have bounded integer coefficients and must thus repeat themselves after finitely many n.”

Thanks to Felix Voigtlaender for corrections.