Let F 1 , . . . , F R be quadratic forms with integer coefficients in n variables. When n ≥ 9R and the variety V (F 1 , . . . , F R ) is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for V (F 1 , . . . , F R ). Previous work in this direction required n to grow at least quadratically with R. We give a similar result for R forms of degree d, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as n > d2 d R. In the case that d ≥ 3, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients. Mathematics Subject Classification

We consider systems F (x) of R homogeneous forms of the same degree d in n variables with integral coefficients. If n ≥ d2 d R + R and the coefficients of F lie in an explicit Zariski open set, we give a nonsingular Hasse principle for the equation F (x) = 0, together with an asymptotic formula for the number of solutions to in integers of bounded height. This improves on the number of variables needed in previous results for general systems F as soon as the number of equations R is at least 2 and the degree d is at least 4.

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of $L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.

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We prove essentially optimal bounds for norms of spectral projectors on thin spherical shells for the Laplacian on the cylinder (R/Z) × R. In contrast to previous investigations into spectral projectors on tori, having one unbounded dimension available permits a compact self-contained proof.