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 → ∞.

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...

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...

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 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.

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 $f:I\to I$ 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 $t=1$.

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.

CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p,q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p,q}\left(B\right)$........ 235. Examples in $L_{p,q}$ theory...................................................................................

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 $x{\left(logx\right)}^{1/\left|G\right|-1}{\left(loglogx\right)}^{{}_k\left(G\right)}$. In this article, it is proved that for every prime p, $₁\left(C_p\oplus C_p\right)=2p$, and it is also proved that $₁\left(C_{mp}\oplus C_{mp}\right)=2mp$ if $₁\left(C_m\oplus C_m\right)=2m$ and m is large enough. In particular, it is shown...

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...

Consider the $n\times n$ matrix with $(i,j)$’th entry $gcd(i,j)$. Its largest eigenvalue ${\lambda }_n$ and sum of entries $s_n$ satisfy ${\lambda }_n>s_n/n$. Because $s_n$ cannot be expressed algebraically as a function of $n$, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that ${\lambda }_n>6{\pi }^{-2}nlogn$ for all $n$. If $n$ is large enough, this follows from F. Balatoni (1969).

Consider a recurrence sequence ${\left(x_k\right)}_{k\in \mathbb{Z}}$ of integers satisfying $x_{k+n}=a_{n-1}{x}_{k+n-1}+...+a₁x_{k+1}+a₀x_k$, where $a₀,a₁,...,a_{n-1}\in \mathbb{Z}$ are fixed and a₀ ∈ -1,1. Assume that $x_k>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k₀}<0$ then for each negative integer -D there exist infinitely many rational primes q such that $q|x_k$ for some k ∈ ℕ and (-D/q) = -1.

Very recently, the generating function $A\left(z\right)$ of the Stern sequence ${\left(a_n\right)}_{n\ge 0}$, defined by $a_0:=0,a_1:=1,$ and $a_{2n}:=a_n,a_{2n+1}:=a_n+a_{n+1}$ for any integer $n\&gt;0$, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A\left(\alpha \right)$ for every algebraic $\alpha$ with $0\&lt;\left|\alpha \right|\&lt;1$, and this result was generalized in [6] to the effect that, for the same $\alpha$’s, all numbers $A\left(\alpha \right),A^{\prime }\left(\alpha \right),A^{\prime \prime }\left(\alpha \right),...$ are algebraically independent. At about the same time, Bacher...

Let $𝒥$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{𝒥}$ satisfy a certain...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

We show a $p$-parity result in a $D_{2p^n}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: $W(E/K,1\oplus \eta \oplus \tau )={(-1)}^{〈1\oplus \eta \oplus \tau ,X_p(E/L)〉}$, where $E$ is an elliptic curve over $K$, $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm{Gal}(L/K)=D_{2p^n}$, and $X_p(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between ${\epsilon }_0$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture...

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.