Let $K$ be a field of degree $n$ over ${\mathbf{Q}}_p$, the field of rational $p$-adic numbers, say with residue degree $f$, ramification index $e$ and differential exponent $d$. Let $O$ be the ring of integers of $K$ and $\mathit{P}$ its unique prime ideal. The trace and norm maps for $K/{\mathbf{Q}}_p$ are denoted $Tr$ and $N$, respectively. Fix $q=p^r$, a power of a prime $p$, and let $\eta$ be a numerical character defined modulo $q$ and of order $o\left(\eta \right)$. The character $\eta$ extends to the ring of $p$-adic integers ${\mathbb{Z}}_p$ in the natural way; namely $\eta \left(u\right)=\eta \left(\tilde{u}\right)$, where $\tilde{u}$ denotes the residue...

For a prime $p$ and positive integers $\ell \<k\<h\<p$ with $d=(h,k,\ell ,p-1)$, we show that $M$, the number of simultaneous solutions $x,y,z,w$ in ${\mathbb{Z}}_p^{*}$ to $x^h+y^h=z^h+w^h$, $x^k+y^k=z^k+w^k$, $x^{\ell }+y^{\ell }=z^{\ell }+w^{\ell }$, satisfies $${\displaystyle M\le 3d^2{(p-1)}^2+25hk\ell (p-1).}$$ When $hk\ell =o\left(p{d}^2\right)$ we obtain a precise asymptotic count on $M$. This leads to the new twisted exponential sum bound $${\displaystyle \left|\sum _{x=1}^{p-1}\chi \left(x\right)e^{2\pi if\left(x\right)/p}\right|\le 3^{\frac{1}{4}}{d}^{\frac{1}{2}}{p}^{\frac{7}{8}}+\sqrt{5}{\left(hk\ell \right)}^{\frac{1}{4}}{p}^{\frac{5}{8}},}$$ for trinomials $f=a{x}^h+b{x}^k+c{x}^{\ell }$, and to results on the average size of such sums.

We consider a function $u:\Omega \to {ℝ}^N$, $\Omega \subset {ℝ}^n$, minimizing the integral ${\int }_{\Omega }\left(\right|D_1{u|}^2+\cdots +|D_{n-1}{u|}^2+|D_{n}u{|}^p)dx$, $2(n+1)/(n+3)\le p<2$, where $D_{i}u=\partial u/\partial x_i$, or some more general functional with the same behaviour; we prove the existence of second weak derivatives $D\left(D_{1}u\right),\cdots ,D\left(D_{n-1}u\right)\in L^2$ and $D\left(D_{n}u\right)\in L^p$.

Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu$ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n𝐱$ for $\mu$-almost all $𝐱\in X^d$ and the behavior of $\mu$ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus ${𝕋}^d$ due to B. Host.