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.
Square-Difference-Free Sets of Size Omega(n^{0.7334...})
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abstract
A set A is square-difference free (henceforth SDF) if there do not exist x,y\in A, x\ne y, such that |x-y| is a square. Let sdf(n) be the size of the largest SDF subset of {1,...,n}. Ruzsa has shown that sdf(n) = \Omega(n^{0.5(1+ \log_{65} 7)}) = \Omega(n^{0.733077...}) We improve on the lower bound by showing sdf(n) = \Omega(n^{0.5(1+ \log_{205} 12)})= \Omega(n^{.7443...}) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers.
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2019 1verdicts
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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.