We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{ 0.7685}$ lacking the configuration $\{ x,x+y,x+y^2\},$ surpassing the $N^{ 3/4}$ limit of Ruzsa's construction for sets lacking a square difference. We also extend Ruzsa's construction to sets lacking polynomial differences for a wide class of univariate polynomials. Finally, we turn to multivariate differences, constructing a set of order $N^{ 1/2}$ lacking a difference equal to a sum of two squares. This is in contrast to the analogous problem of sets lacking a difference equal to a prime minus one, where the current record is of order $N^{ o(1)}.$
Categories: math.NT math.CO