Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow \mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f} \Vert_1 \leq 1$. Then $\min_{x \in \Zp} |f(x)| = O(\log p)^{-1/3 + \epsilon}$. One should think of $f$ as being ``approximately continuous''; our result is then an ``approximate intermediate value theorem''. As an immediate consequence we show that if $A \subseteq \Zp$ is a set of cardinality $\lfloor p/2\rfloor$, then $\sum_r |\widehat{1_A}(r)| \gg (\log p)^{1/3 - \epsilon}$. This gives a result on a ``mod $p$'' analogue of Littlewood's well-known problem concerning the smallest possible $L^1$-norm of the Fourier transform of a set of $n$ integers. Another application is to answer a question of Gowers. If $A \subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is some $x \in \Zp$ such that \[ | |A \cap (A + x)| - p/4 | = o(p).\]

We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product $H_{f}H_g^\ast$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products $H_{g}T_{\bar{f}}$ and $T_{f}H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.

Let $q$ be an algebraic integer of degree $d \geq 2$. Consider the rank of the multiplicative subgroup of $\BbC^*$ generated by the conjugates of $q$. We say $q$ is of {\em full rank} if either the rank is $d-1$ and $q$ has norm $\pm 1$, or the rank is $d$. In this paper we study some properties of $\BbZ[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results. \begin{compactenum}[\rm(1)] \item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$. \item If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\BbZ[q^n] = \BbZ[r^n]$ for infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$, where $r'$ is some conjugate of $r$ and $\omega$ is some root of unity. \item Let $r$ be an algebraic integer of degree at most $3$. Then there are at most $40$ Pisot numbers $q$ such that $\BbZ[q] = \BbZ[r]$. \item There are only finitely many Pisot-cyclotomic numbers of any fixed order. \end{compactenum}

For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form $\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we show that if the mean curvature vector of $M^n$ is parallel and the sectional curvature $K$ of $M^n$ satisfies some inequality, then the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is parallel and our manifold $M^n$ is a space form.