Let $\alpha \>1$ be irrational. Several authors studied the numbers $${\ell }^m\left(\alpha \right)=inf\left\{\right|y|:y\in {\Lambda }_m,y\ne 0\},$$ where $m$ is a positive integer and ${\Lambda }_m$ denotes the set of all real numbers of the form $y={ϵ}_0{\alpha }^n+{ϵ}_1{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_n$ with restricted integer coefficients $|{ϵ}_i|\le m$. The value of ${\ell }^1\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell }^m\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell }^m\left(\alpha \right)$ is determined for irrational numbers $\alpha$, satisfying ${\alpha }^2=a\alpha \pm 1$ with a positive integer $a$.

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

We consider digit expansions ${\sum }_{j=0}^{\ell -1}{\Phi }^j\left(d_j\right)$ with an endomorphism $\Phi$ of an Abelian group. In such a numeral system, the $w$-NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$-NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

For an algebraic number field $K$ with $3$-class group ${\mathrm{Cl}}_3\left(K\right)$ of type $(3,3)$, the structure of the $3$-class groups ${\mathrm{Cl}}_3\left(N_i\right)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group ${\mathrm{G}}_3^2\left(K\right)=\mathrm{Gal}\left({\mathrm{F}}_3^2\left(K\right)\right|K)$ of the second Hilbert $3$-class field ${\mathrm{F}}_3^2\left(K\right)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}\left(\sqrt{D}\right)$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,...,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,...,N_4$. This...

Let $F$ be an imaginary quadratic field and $𝒪$ its ring of integers. Let $𝔫\subset 𝒪$ be a non-zero ideal and let $p\>5$ be a rational inert prime in $F$ and coprime with $𝔫$. Let $V$ be an irreducible finite dimensional representation of ${\overline{𝔽}}_p\left[{\mathrm{GL}}_2\left({𝔽}_{p^2}\right)\right]$. We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\overline{𝔽}}_p\otimes de{t}^e$ for some $e\ge 0$; except possibly in some few cases.

If the space $𝒬$ of quadratic forms in ${ℝ}^n$ is splitted in a direct sum ${𝒬}_1\oplus ...\oplus {𝒬}_k$ and if $X$ and $Y$ are independent random variables of ${ℝ}^n$, assume that there exist a real number $a$ such that $E\left(X\right|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E\left(q\left(X\right)\right|X+Y)=b_{i}q(X+Y)$ for any $q$ in ${𝒬}_i.$ We prove that this happens only when $k=2$, when ${ℝ}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...