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We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields. 

We provide a simple criterion on a family of functions that implies a square function estimate on $L^p$for every even integer $p\ge <\#comment/>2$. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known. 

Braverman and Kazhdan proposed a conjecture, later refined by Ngô and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B. Liu and later the first two authors proved these conjectures for certain spherical varieties Y built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on Y: We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on the affine closures of Braverman–Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory.