Higher Order Asymptotic Formulas for Traces of Toeplitz Matrices with Symbols in Hölder-Zygmund Spaces

On unimodular multilinear forms with small norms on sequence spaces

The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:\ell_{p_{1}}^{n}\times \cdots\times\ell_{p_{m}}^{n}\longrightarrow\mathbb{R}$ (or $\mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack2,\infty]$ and when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the …