Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system ${{\varphi }_d\left(x\right)=M^{-1}(x+d)}_{d\in D}$ has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that $D=M^{\gamma }D₀$, where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results...

Let A be a d × d integral expanding matrix and let $S_j\left(x\right)=A^{-1}(x+d_j)$ for some $d_j\in {\mathbb{Z}}^d$, j = 1,...,m. The iterated function system (IFS) ${S_j}_{j=1}^m$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof...

We reduce the problem of describing all $ℳf_m$-natural operators  transforming general affine connections on $m$-manifolds into general affine ones to the known description of all $GL\left({𝐑}^m\right)$-invariant maps ${𝐑}^{m*}\otimes {𝐑}^m\to {\otimes }^k{𝐑}^{m*}\otimes {\otimes }^k{𝐑}^m$ for $k=1,3$.

Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if $\left|detM\right|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set ${}_M$ of points in $A_M$ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of ${}_M$ is positive. For this special class the full description of ${}_M$ is given as well. This paper continues our...

A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $\Phi \left(S\right)={\bigcup }_{1\le j\le t}{\varphi }_j\left(S\right)$. If Φ(S) ⊂ S let $E_{\Phi }\left(S\right)$ denote ${\bigcap }_{k\ge 0}{\Phi }^k\left(S\right)$. For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{\Phi }\left(S\right)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also...

Let $X_i$, i∈ I, and $Y_j$, j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of ${\prod }_{i\in I}{X}_i$ onto ${\prod }_{j\in J}{Y}_j$. We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_i$ onto $Y_{b\left(i\right)}$, i∈ I.

A measure is called $L^p$-improving if it acts by convolution as a bounded operator from $L^q$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^p$-improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^p$-improving.

Let d be a positive integer and μ a generalized Cantor measure satisfying $\mu ={\sum }_{j=1}^m{a}_j\mu \circ S_j^{-1}$, where $0<a_j<1$, ${\sum }_{j=1}^m{a}_j=1$, $S_j=\rho R+b_j$ with 0 < ρ < 1 and R an orthogonal transformation of ${ℝ}^d$. Then ⎧1 < p ≤ 2 ⇒ ⎨$su{p}_{r>0}{r}^{d(1/{\alpha }^{\text{'}}-1/p^{\text{'}})}({\int }_{J_x^r}{\left|\mu ̂\left(y\right)\right|}^{p^{\text{'}}}{dy)}^{1/p^{\text{'}}}\le D₁{\rho }^{-d/{\alpha }^{\text{'}}}$, $x\in {ℝ}^d$, ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’$,$where $J_x^r={\prod }_{i=1}^d(x_i-r/2,x_i+r/2)$, α’ is defined by ${\rho }^{d/{\alpha }^{\text{'}}}={\left({\sum }_{j=1}^m{a}_j^p\right)}^{1/p}$ and the constants D₁ and D₂ depend only on d and p.

We determine the duals of the homogeneous matrix-weighted Besov spaces ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ which were previously defined in [5]. If W is a matrix $A_p$ weight, then the dual of ${Ḃ}_p^{\alpha q}\left(W\right)$ can be identified with ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$ and, similarly, $\left[{ḃ}_p^{\alpha q}\left(W\right)\right]*\approx {ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$. Moreover, for certain W which may not be in the $A_p$ class, the duals of ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ are determined and expressed in terms of the Besov spaces ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$ and ${ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$, which we define in terms of reducing operators ${A_Q}_Q$ associated with W. We also develop the basic theory of these reducing operator Besov spaces....

Let $S_i:{ℝ}^d\to {ℝ}^d$ for i = 1,..., N be contracting similarities, let $(p₁,...,p_N,p)$ be a probability vector and let ν be a probability measure on ${ℝ}^d$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on ${ℝ}^d$ such that $\mu ={\sum }_{i=1}^N{p}_i\mu \circ S_i^{-1}+p\nu$. We give satisfactory estimates for the lower and upper bounds of the $L^q$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

Let $G:=\mathrm{SO}{(n,1)}^{\circ }$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{ℬ}_T\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {ℬ}_T=ℳ\left({ℬ}_T\right)+O\left(ℳ{\left({ℬ}_T\right)}^{1-\eta }\right)$ for an explicit measure $ℳ$ on $H\setminus G$ which depends on $\Gamma$. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma$ in arithmetic...

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that $H^1(G,\pi )=0$ for every isometric representation $\pi$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we...

We relate centroaffine immersions $f:Mⁿ\to {ℝ}^{n+1}$ to horizontal immersions g of Mⁿ into $S_{n+1}^{2n+1}\left(1\right)$ or $H_n^{2n+1}(-1)$. We also show that f is an equiaffine sphere, i.e. the centroaffine normal is a constant multiple of the Blaschke normal, if and only if g is minimal.

Associated to an Hadamard matrix $H\in M_N\left(ℂ\right)$ is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with $G\subset S{⁺}_N$. We study a certain family of discrete measures ${\mu }^r\in [0,N]$, coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type ${\int }_0^N{(x/N)}^{p}d{\mu }^r\left(x\right)={\int }_0^N{(x/N)}^{r}d{\nu }^p\left(x\right)$, where ${\mu }^r,{\nu }^r$ are the truncations of the spectral measures μ,ν associated to $H,H^t$. We also prove, using these truncations ${\mu }^r,{\nu }^r$, that for any deformed Fourier matrix $H=F_M{\otimes }_Q{F}_N$ we have μ = ν.

We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_1,\cdots ,u_m):U\to {ℝ}^m$ such that $U\cap X=\{u_1\cdots u_r=0\}$, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash...

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\text{'}}$ (the magnetization) of two variables $x^{\text{'}}$: $E_{\epsilon }\left(m^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·m^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·m^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be $S^1$-valued maps $m^{\text{'}}$ of vanishing distributional divergence ${\nabla }^{\text{'}}·m^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel...

Let $f,g:U\to {𝔸}^1$ be two regular functions from the smooth affine complex variety $U$ to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system ${𝒪}_U{e}^g$ with respect to $f$. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.

Let $S$ be a fixed symmetric finite subset of $S{L}_d\left({𝒪}_K\right)$ that generates a Zariski dense subgroup of $S{L}_d\left({𝒪}_K\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({𝒪}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.

Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building ${𝔅}^1\left(H\right)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building ${𝔅}^1\left(G\right)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{\prime }$ be maps from ${𝔅}^1\left(H\right)$ to ${𝔅}^1\left(G\right)$ which preserve the Moy–Prasad filtrations....

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

For a double complex $(A,d^{\text{'}},d^{\text{'}\text{'}})$, we show that if it satisfies the $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma and the spectral sequence $\left\{E_r^{p,q}\right\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma.

Let $\mathbb{Z}$ be the set of integers, ${\mathbb{N}}_0$ the set of nonnegative integers and $F(x_1,x_2)=u_1{x}_1+u_2{x}_2$ be a binary linear form whose coefficients $u_1$, $u_2$ are nonzero, relatively prime integers such that $u_1{u}_2\ne \pm 1$ and $u_1{u}_2\ne -2$. Let $f:\mathbb{Z}\to {\mathbb{N}}_0\cup \left\{\infty \right\}$ be any function such that the set $f^{-1}\left(0\right)$ has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set $A$ of integers such that $r_{A,F}\left(n\right)=f\left(n\right)$ for all integers $n$, where $r_{A,F}\left(n\right)=\left|\{(a,a^{\text{'}}):n=u_{1}a+u_2{a}^{\text{'}}:a,a^{\text{'}}\in A\}\right|$. We add the structure of difference for the binary linear form $F(x_1,x_2)$.

We study the group $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ of tame automorphisms of a smooth affine $3$-dimensional quadric, which we can view as the underlying variety of ${\mathrm{SL}}_2\left(ℂ\right)$. We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $\mathrm{CAT}\left(0\right)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ is linearizable, and that $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ satisfies the Tits alternative.

For the dynamics $x^{\text{'}\text{'}}=-{\nabla }_{x}V\left(x\right)$, an equilibrium point $\underline{x}$ are always unstable when on a neighborhood of $\underline{x}$ the non constant $V$ satisfies $P(x,\partial )V=0$ for a real second order elliptic $P$. The proof uses a result of Kozlov [6].

We consider elementary operators $x\mapsto {\sum }_{j=1}^n{a}_{j}x{b}_j$, acting on a unital Banach algebra, where $a_j$ and $b_j$ are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families $a_j$ and $b_j$, i.e. $a_j=a_j^{\text{'}}+i{a}_j^{\text{'}\text{'}}$ ($b_j=b_j^{\text{'}}+i{b}_j^{\text{'}\text{'}}$), where all $a_j^{\text{'}}$ and $a_j^{\text{'}\text{'}}$ ($b_j^{\text{'}}$ and $b_j^{\text{'}\text{'}}$) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class...

We present a new theorem on the differential inequality $u^{\left(m\right)}\le w\left(u\right)$. Next, we apply this result to obtain existence theorems for the equation $x^{\left(m\right)}=f(t,x)$.