A refinement of Cauchy-Schwarz complexity

We introduce a notion of complexity for systems of linear forms called sequential Cauchy-Schwarz complexity, which is parametrized by two positive integers $k,\ell$ and refines the notion of Cauchy-Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy-Schwarz complexity at most $(k,\ell)$ then any average of 1-bounded functions over this system is controlled by the $2^{1-\ell}$-th power of the Gowers $U^{k+1}$-norms of the functions. For $\ell=1$ this agrees with Cauchy-Schwarz complexity, but for $\ell>1$ there are families of systems that have sequential Cauchy-Schwarz complexity at most $(k,\ell)$ whereas their Cauchy-Schwarz complexity is greater than $k$. For instance, for $p$ prime and $k\in \mathbb{N}$, the system of forms $\big\{\phi_{z_1,z_2}(x,t_1,t_2)= x+z_1 t_1+z_2t_2\;|\; z_1,z_2\in [0,p-1], z_1+z_2

Keywords:

• Mathematics
• Ergodic theory
• Vector space
• Prime (order theory)
• Omega
• Polynomial (hyperelastic model)
• Cauchy–Schwarz inequality
• Inverse
• Combinatorics

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