Sharp estimates for Gowers norms on discrete cubes

We study optimal dimensionless inequalities $$\mathrm{\parallel }f{\mathrm{\parallel }}_{U^k}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathrm{\ell }}^{p_{k,n}}}$$ that hold for all functions $f\text{\hspace{0.17em}⁣}:{\mathbb{Z}}^d\to \mathbb{C}$ supported in $\{0,1,\dots ,n-1{\}}^d$ and estimates $$\mathrm{\parallel }{1}_A{\mathrm{\parallel }}_{U^k}^{2^k}\le \mathrm{\mid }A{\mathrm{\mid }}^{t_{k,n}}$$ that hold for all subsets $A$ of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n}{t}_{k,n}=2^k$. This is used to prove the three main results of the paper: an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao, two-sided asymptotic estimates for $t_{k,n}$ as $n\to \mathrm{\infty }$ for a fixed $k\ge 2$, which generalize a theorem by Shao, and a precise asymptotic formula for $t_{k,n}$ as $k\to \mathrm{\infty }$ for a fixed $n\ge 2$.