This is a monograph that draws on a wide range of mathematics but applies the results to very specific problems regarding the existence of arithmetic progressions in subsets of the integers.
The classic and first result in this field is Roth’s 1953 theorem that a subset of the integers with positive density contains an infinity of three-term arithmetic progressions. The main applications given in this book are Roth’s theorem and the analogous result for the primes (the primes have density 0 in the integers, so Roth’s result does not apply directly). In 2004 Tao and Ben Green went further and proved that there are arbitrarily long arithmetic progressions in the primes. That result is not proved in the present book, but it depends on tools developed here.
The phrase “higher order” in the title refers to the degree of the polynomial describing the phase shift between consecutive basis elements. Traditional Fourier analysis uses a linear shift, so the multiplier in the argument of the trigonometric function or complex exponential is a linear function of the index. “Higher order” methods use quadratic or higher degree polynomials. The linear methods are adequate for results about three-term arithmetic progressions, but not for longer ones.
The book uses a dazzling array of methods from many areas of mathematics, including classical analysis, Fourier analysis on finite groups, measure theory, probability, equidistribution, dynamical systems, and cohomology. The book also sketches some alternate approaches using nonstandard analysis and ultrafilters.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.