We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets A ⊂ Z and show quantitative bounds on the size of A if there are no non-trivial solutions. We use the circle method and Roth's density increment argument. Due to a restriction theory approach we can deal with equations in s ≥ 7 variables.

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Abstract. We investigate the size of L p -integrals for exponential sums over k-free numbers and prove essentially tight bounds.