The inequality $ʃlog|\sum a_n{e}^{2\pi i{\phi }_n}|d{\phi }_1\dots d{\phi }_n\ge Clog\left(\sum \right|a_n{{|}^2)}^{1/2}$ with an absolute constant C, and similar ones, are extended to the case of $a_n$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2\pi i\phi }$.

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of ${\mathbb{Z}}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $\left|A\right|\sim logq$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll logq$ and $logk\ll logq$, provided the...

For the real quadratic map $P_a\left(x\right)=x^2+a$ and a given $ϵ\>0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_a^n{|}_J$ univalent, with bounded distortion and $B(0,ϵ)\subseteq P_a^n\left(J\right)$ for some $n\in \mathbb{N}$. The $ϵ$-weakly expanding set is the set of points which do not have good expansion properties. Let $\alpha$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[\alpha ,-\alpha ]$. We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

Let $(X_i,i=1,2,...)$ be the normalized gaussian system such that $X_i\in N(0,1)$, i = 1,2,... and let the correlation matrix ${\rho }_{ij}=E\left(X_i{X}_j\right)$ satisfy the following hypothesis: $C=su{p}_{i\ge 1}{\sum }_{j=1}^{\infty }\left|{\rho }_{i,j}\right|<\infty$. We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site...

We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{{log}^{\alpha }}^2$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{\alpha -1}}^2$, while if $\alpha >2$ and $0<\epsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{{log}^{\alpha }}^2$ into $A_{{log}^{\alpha -2-\epsilon }}^2$.We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space ${ℬ}_{{log}^{\alpha }}$, $\alpha \in ℝ$, into itself, while $H$ maps ${ℬ}_{{log}^{\alpha }}$ into ${ℬ}_{{log}^{\alpha +1}}$.In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space ${ℬ}_{{log}^{\alpha }}^1$, $\alpha >0$, into ${ℬ}_{{log}^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps ${ℬ}_{{log}^{\alpha }}^1$, $\alpha \ge 0$,...

Let ${𝒯}_n$ denote the set of log canonical thresholds of pairs $(X,Y)$, with $X$ a nonsingular variety of dimension $n$, and $Y$ a nonempty closed subscheme of $X$. Using non-standard methods, we show that every limit of a decreasing sequence in ${𝒯}_n$ lies in ${𝒯}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that ${𝒯}_n$ is closed in $𝐑$; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

Let $\left[x\right]$ be an integer part of $x$ and $d\left(n\right)$ be the number of positive divisor of $n$. Inspired by some results of M. Jutila (1987), we prove that for $1<c<\frac{6}{5}$, $$\sum _{n\le x}d\left(\left[n^c\right]\right)=cxlogx+(2\gamma -c)x+O\left(\frac{x}{logx}\right),$$ where $\gamma$ is the Euler constant and $\left[n^c\right]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

Let $$T\left(q\right)=\sum _{k=1}^{\infty }d\left(k\right)q^k,\left|q\right|<1,$$ where $d\left(k\right)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we prove that $$H\left(q\right)=T\left(q\right)-\frac{log(1-q)}{log\left(q\right)}$$ is strictly increasing on $(0,1)$, while $$F\left(q\right)=\frac{1-q}{q}H\left(q\right)$$ is strictly decreasing on $(0,1)$. These results are then applied to obtain various inequalities, one of which states that the double inequality $$\alpha \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)}<T\left(q\right)<\beta \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)},0<q<1,$$ holds with the best possible constant factors $\alpha =\gamma$ and $\beta =1$. Here, $\gamma$ denotes Euler’s constant. This refines a result of Salem, who...

Let K be an algebraic number field with non-trivial class group G and ${}_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_k\left(x\right)$ denote the number of non-zero principal ideals $a_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_k\left(x\right)$ behaves for x → ∞ asymptotically like $x{\left(logx\right)}^{1-1/\left|G\right|}{\left(loglogx\right)}^{{}_k\left(G\right)}$. We prove, among other results, that $₁\left(C_{n₁}\oplus C_{n₂}\right)=n₁+n₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

Let ${ℳ}^d$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in ${ℳ}^d$. Let us fix a Lebesgue measure $V_B$ in ${ℳ}^d$ with $V_B\left(B\right)=1$. This measure will play the role of the volume in ${ℳ}^d$. We consider an arbitrary simplex T in ${ℳ}^d$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_B\left(T\right)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_B\left(T\right)$ is zero.

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

For any positive integer k and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ and $r_{1,l}(A,n)=r_{1,l}(\mathbb{N}\setminus A,n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all n ≥ n₀, we have $r_{1,k}(A,n)\to \infty$ as n → ∞.

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.

We give some characterizations of the class ${}_m\left(\Omega \right)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

Let $X$ be a complex manifold with strongly pseudoconvex boundary $M$. If $\psi$ is a defining function for $M$, then $-log\psi$ is plurisubharmonic on a neighborhood of $M$ in $X$, and the (real) 2-form $\sigma =i\partial \overline{\partial }(-log\psi )$ is a symplectic structure on the complement of $M$ in a neighborhood of $M$ in $X$; it blows up along $M$. The Poisson structure obtained by inverting $\sigma$ extends smoothly across $M$ and determines a contact structure on $M$ which is the same as the one induced by the complex structure. When $M$ is compact, the Poisson structure...

We prove the $L^p$ boundedness of the Marcinkiewicz integral operators ${\mu }_{\Omega }$ on ${ℝ}^{n₁}\times \cdots \times {ℝ}^{n_k}$ under the condition that $\Omega \in L{\left(logL\right)}^{k/2}{(}^{n₁-1}\times \cdots {\times }^{n_k-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element ${𝔽}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{\mathrm{an}}$. Moreover, using the scheme-theoretic...

We prove that for a normed linear space $X$, if $f_1:X\to ℝ$ is continuous and semiconvex with modulus $\omega$, $f_2:X\to ℝ$ is continuous and semiconcave with modulus $\omega$ and $f_1\le f_2$, then there exists $f\in C^{1,\omega }\left(X\right)$ such that $f_1\le f\le f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega \left(t\right)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C_{loc}^{1,\omega }$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Let $k$ be a field of characteristic $p\&gt;0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...