A digit-construction gives subsets of [N] of size N^{0.7685} with no {x, x+y, x+y^2}, breaking the N^{3/4} barrier for Ruzsa square-difference sets, plus polynomial lower bounds for multivariate differences.
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Lower bounds in the polynomial Szemer\'edi theorem
A digit-construction gives subsets of [N] of size N^{0.7685} with no {x, x+y, x+y^2}, breaking the N^{3/4} barrier for Ruzsa square-difference sets, plus polynomial lower bounds for multivariate differences.