For any integer polynomial P with P(0)=0, any subset of [N] avoiding x, x+P(y), ..., x+kP(y) has size at most N (log log log N)^{-c}, with stronger bounds when P'(0)!=0.
Efficient Equidistribution of Nilsequences
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abstract
We give improved bounds for the equidistribution of (multiparameter) nilsequences subject to any degree filtration. The bounds we obtain are single exponential in dimension, improving on double exponential bounds of Green and Tao. To obtain these bounds, we overcome "induction of dimension'' which is ubiquitous throughout higher order Fourier analysis. The improved equidistribution theory is a crucial ingredient in the quasi-polynomial $U^4[N]$ inverse theorem of the author and its extension to the quasi-polynomial $U^{s + 1}[N]$ inverse theorem in joint work with Sah and Sawhney. These results lead to further applications in combinatorial number theory such as bounds for linear equations in the primes which save an arbitrary power of logarithm, which match the bounds Vinogradov obtained for the odd Goldbach conjecture.
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On polynomial progressions via transference
For any integer polynomial P with P(0)=0, any subset of [N] avoiding x, x+P(y), ..., x+kP(y) has size at most N (log log log N)^{-c}, with stronger bounds when P'(0)!=0.