We obtain nontrivial exponents for Erdős-Falconer type point configuration problems. Let T k (E) denote the set of distinct congruent k-dimensional simplices determined by (k + 1)-Results of this type were previously obtained for triangles in the plane (k = d = 2) in [9] and for higher k and d in [8]. We improve upon those exponents, using a group action perspective, which also sheds light on the classical approach to the Falconer distance problem.

We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving .k C 1/-point configurations in geometric measure theory, with k 2, including the distribution of simplices, volumes and angles determined by the points of fractal subsets E R d , d 2. If T k .E/ denotes the set of noncongruent .k C 1/-point configurations determined by E, we show that if the Hausdorff dimension of E is greater than d d 1 2k , then the kC1 2 -dimensional Lebesgue measure of T k .E/ is positive. This complements previous work on the Falconer conjecture ([5] and the references there), as well as work on finite point configurations [6,10]. We also give applications to Erdős-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in [7].

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Abstract. For functions F, G on R n , any k-dimensional affine subspace H ⊂ R n , 1 ≤ k < n, and exponents p, q, r ≥ 2 with 1 p + 1 q + 1 r = 1, we prove the estimateHere, the mixed norms on the right are defined in terms of the Fourier transform byLebesgue measure on the affine subspace H ⊥ ξ := ξ + H ⊥ . Dually, one obtains restriction theorems for the Fourier transform on affine subspaces. We use this, and a maximal variant, to prove results for a variety of multilinear convolution operators, including L p -improving bounds for measures; bilinear variants of Stein's spherical maximal theorem; estimates for m-linear oscillatory integral operators; Sobolev trace inequalities; and bilinear estimates for solutions to the wave equation.

We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions d⩾5. That is, we show that this operator is bounded on lpfalse(Zdfalse)×lqfalse(Zdfalse)→lrfalse(Zdfalse) for 1/p+1/q⩾1/r and r>d/(d−2) and we show this range is sharp. Our approach mirrors that used by Jeong and Lee in the continuous setting. For dimensions d=3,4, our previous work, which used different techniques, still gives the best known bounds. We also prove analogous results for higher degree k, ℓ‐linear operators.