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We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve in flat tori are universal under various families of randomness.

Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.

Given a smooth integer q, we use existing upper bounds for character sums to find a lower bound for the size of a multiplicative subgroup of the integers modulo q which contains the image of an interval of consecutive integers I⊂Zq under a polynomial f∈Z[X].

We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $\lambda^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed., Comment: Version 2, with Mei-Chu Chang as a co-author, extends the range in which Version 1 worked

We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively.

We use bounds of mixed character sums modulo a square-free integer q of a special structure to estimate the density of integer points on the hypersurface $$\begin{aligned} f_1(x_1) + \cdots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} \end{aligned}$$ for some polynomials $$f_i \in {\mathbb {Z}}[X]$$ and nonzero integers a and $$k_i$$ , $$i=1, \ldots , n$$ . In the case of $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^2\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$ the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^n\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$ it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.

The dust acoustic wave (DAW), associated with longitudinal dust oscillations in dusty plasmas, can be self-excited from the free energy of ion streaming. It is not only a fundamental plasma wave but also a paradigm to understand the generic dynamical behaviors of self-excited nonlinear longitudinal density waves through optically monitoring particle motion and dust density evolutions over a large area. In this paper, the dynamical behaviors of the wave-particle interaction and wave breaking in ordered self-excited DAW with straight wave fronts, and the defect-mediated wave turbulence with fluctuating defects and chaotic low amplitude hole filaments along defect trajectories in the 2+1D space-time space, are briefly reviewed. The first experimental observation of acoustic vortices with helical waveforms in self-excited acoustic-type defect-mediated wave turbulence, and the dynamics of spontaneous pair generation, propagation, and pair annihilation of acoustic vortices, is demonstrated and discussed.

Let A , B be invertible n × n matrices over the finite field F q with irreducible characteristic polynomials. For k ∈ Z + , denote M k ( A , B ) : = { f ( A ) g ( B ) : f , g ∈ F q [ x ] , with deg f , deg g k } . Assume q ⩾ 2 n , we prove that | M k ( A , B ) | > 4 − k q min ( n , 2 k − 1 ) . Moreover, let d = dim Ker ( A B − B A ) , we prove | M k ( A , B ) | > 1 16 ( n d ) ( 2 ( n − d ) ) n − d 2 q k + min ( k 2 , n − d 2 ) .

We prove the following analogue of Silverman's results [9] for pairs of maps. Let d ≥ 2 be an integer, K / Q a number field, and N = N K / Q ( P ) the norm of an ideal P ⊂ O K . Let h ( z ) ∈ K [ z ] be non-constant and not of the form h ( z ) = ξ z , ξ d − 1 = 1 . Denote f t ( z ) = z d + t , g t ( z ) = z d + h ( t ) , and F ( l ) the l-th iteration of F. There are constants c 1 , c 2 depending on d and h such that the following holds. For almost all prime ideals P ⊂ O K , there is a finite subset T ⊂ F ¯ P , | T | ≤ c 1 such that if t ∈ F ¯ P ∖ T at least one of the sets (1) { f t ( l ) ( 0 ) : l = 1 , 2 , ⋯ , [ c 2 log ⁡ N ] } , { g t ( l ) ( 0 ) : l = 1 , 2 , ⋯ , [ c 2 log ⁡ N ] } consists of distinct elements.

In this survey we review some old and new results on character sums and their applications to various problems in analytic number theory, e.g., smallest quadratic nonresidues, Dirichlet L-function, Linnik’s problem. We also discuss open problems in this area. Some of the techniques involved belong to arithmetic combinatorics. One may hope for these methods to lead to further progress.

Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-\kappa}$ and $r$ an arbitrarily fixed positive integer. Assuming $\kappa=\kappa(r)>0$ and $p$ large enough, it is shown that any proportional subset $A\subset G$ contains non-trivial arithmetic progressions of length $r$. The main ingredient is the Szemer\'{e}di-Green-Tao theorem.

In this work, we establish a nontrivial level of distribution for densities on $\{1,\ldots, N\}$ obtained by a biased coin convolution. As a consequence of sieving theory, one then derives the expected lower bound for the weight of such densities on sets of almost primes.

We establish average bounds on the partial quotients of fractions b/p, with p prime, b taken in a multiplicative subgroup of (Z/pZ)⁎ and for “most” primitive elements b. Our result improves upon earlier work due to G. Larcher. The behavior of the partial quotients of b/p is well known to be crucial to the statistical properties of the pseudo-congruential number generator (modp). As a corollary, estimates on their pair correlation are refined.

We establish character sum bounds of the form | ∑ a ⩽ x ⩽ a + H b ⩽ y ⩽ b + H χ ( x 2 + k y 2 ) | p − τ H 2 , where χ is a non-trivial character (mod p), p 1 4 + e H p , and | a | , | b | p e / 2 H . As an application, we obtain that given k ∈ Z ∖ { 0 } , x 2 + k is a quadratic non-residue (mod p) for some 1 ⩽ x p 1 2 e + ϵ .

Assuming the Weak Polynomial Freiman–Ruzsa Conjecture, we derive some consequences on sum-products and the growth of subsets of SL 3 ( C ) . To cite this article: M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

In this paper we prove the following theorems in incidence geometry. \proclaim{1} There is $\delta >0$ such that for any $P_1, \cdots, P_4$, and $Q_1, \cdots, Q_n \in \Bbb C^2,$ if there are $\leq n^{\frac {1+\delta}2}$ 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 $ then the cross ratio of the four points is algebraic.\endproclaim \proclaim{2} Given $c>0$, there is $\delta 0$ such that for any $P_1, P_2, P_3$ noncollinear, and $Q_1, \cdots, Q_n \in \Bbb C^2$, if there are $\leq c n^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i,j$, then for any $P\in \Bbb C^2\smallsetminus\{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$.\endproclaim \proclaim{3} Given $c>0$, there is $\epsilon 0$ such that for any $P_1, P_2, P_3$ collinear, and $Q_1, \cdots, Q_n \in \Bbb C^2$ (respectively, $\Bbb R^2$), if there are $\leq c n^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i,j$, then for any $P$ not lying on the line $L(P_1,P_2)$, we have at least $n^{1-\epsilon}$ (resp. $ n/\log n$) distinct lines between $P$ and $Q_j$.\endproclaim The main ingredients used are the subspace theorem, Balog-Szemer\'edi-Gowers Theorem, and Szemer\'edi-Trotter Theorem. We also generalize the theorems to high dimensions, extend Theorem 1 to $\Bbb F_p^2$, and give the version of Theorem 2 over $\Bbb Q$.

In this note we consider some quantitative versions of conjectures made by Arnold related to Galois dynamics in finite fields. We refine some results by Shparlinski using exponential sum results.

We establish bounds on exponential sums ∑ x ∈ F q ψ ( x n ) where q = p m , p prime, and ψ an additive character on F q . They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order ( m ⩾ 2 ) . More precisely, a non-trivial estimate is obtained provided n satisfies gcd ( n , q − 1 p ν − 1 ) p − ν q 1 − e for all 1 ⩽ ν m , ν | m , where e > 0 is arbitrary. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

In this paper we extend the exponential sum results from [BK] and [BGK] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size qδ, for any given δ > 0. The method consists in first establishing a ‘sumproduct theorem’ for general subsets A of $$\mathbb{Z}^{q} $$ . If q is prime, the statement, proven in [BKT], expresses simply that either the sum-set A + A or the product-set A.A is significantly larger than A, unless |A| is near q. For composite q, the presence of nontrivial subrings requires a more complicated dichotomy, which is established here. With this sum-product theorem at hand, the methods from [BGK] may then be adapted to the present context with composite moduli. They rely essentially on harmonic analysis and graph-theoretical results such as Gowers’ quantitative version of the Balog–Szemeredi theorem. As a corollary, we get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p r (p prime) for all r. Only the case r = 2 has been treated earlier in works of Heath-Brown and Heath-Brown and Konyagin (using Stepanov’s method). We also get exponential sum estimates for (possibly incomplete) sums involving exponential functions, as considered for instance in [KS].

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 $\log |A| \sim \log q$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll \log q$ and $\log k \ll \log q$, provided the residue sets $\pi_{q'} (A)$ are large for all large divisors $q'$ of $q$. Even in the special case of prime modulus $q$, some results are new, when considering large but bounded sets $A$. It follows for instance from our estimates that one can obtain $r$ as small as $r\sim\xfrac {\log q}{\log|A|}$ with similar restriction on $k$, something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking for $A=H$ a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on $\min_{a \in \mathbb Z_q^*} \max_{x \in H} \Vert\xfrac {ax}q\Vert $ are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of $x^m = a $ (mod $q)$ to composite modulus $q$.

We apply the refined composite multiscale entropy (MSE) method to a one-dimensional directed small-world network composed of nodes whose states are binary and whose dynamics obey the majority rule. We find that the resulting fluctuating signal becomes dynamically complex. This dynamical complexity is caused (i) by the presence of both short-range connections and long-range shortcuts and (ii) by how well the system can adapt to the noisy environment. By tuning the adaptability of the environment and the long-range shortcuts we can increase or decrease the dynamical complexity, thereby modeling trends found in the MSE of a healthy human heart rate in different physiological states. When the shortcut and adaptability values increase, the complexity in the system dynamics becomes uncorrelated.

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences f(x) ≡ y (mod p) and f(x) ≡ y (mod p), with a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We give two applications of these results to counting hyperelliptic curves in isomorphism classes modulo p and to the diameter of partial trajectories of a polynomial dynamical system modulo p.

LetA be a finite subset of some normed division algebra over ℝ with cardinality ⋎A⋎. We prove that either the sum set or the product set ofA has cardinality ⋎A⋎1+δ for some δ>0.

In this Note, we extend the results of Bourgain, Konyagin and Glibichuk to certain composite moduli q involving few ‘large’ primes. First a ‘sum-product’ theorem for subsets A of Zq is obtained, ensuring that |A+A|+|A.A|>c|A|1+ɛ provided |A

The following analogue of the Erdos–Szemeredi sum-product theorem is shown. Let A=f1,⋯,fN be a finite set of N arbitrary distinct functions on some set. Then either the sum set fi+fj or the product set fifj has at least N1+c elements, where c>0 is an absolute constant. We use Freiman's lemma and Balog–Szemeredi–Gowers Theorem on graphs and combinatorics. As a corollary, we obtain an Erdos–Szemeredi type theorem for semi-simple commutative Banach algebras R. Thus if A⊂R is a finite set, |A| large enough, then |A+A|+|A.A|>|A|1+c, where c>0 is an absolute constant. The result and method have various consequences, for instance decay estimates on the convolution powers of finite multiplication subgroups. Let H be a finite multiplicative subgroup of R (as above) and let N=|H|, v=1N∑x∈Hδx. Then, for all constant c, there is a k=k(c) such that maxz∈Rv(k)(z)

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erd��s and Szemer��di on sum and product sets., 33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)

Let A⊂Z be a finite set of integers of cardinality |A|=N⩾2. Given a positive integer k, denote kA (resp. A(k)) the set of all sums (resp. products) of k elements of A. We prove that for all b>1, there exists k=k(b) such that max(|kA|,|A(k)|)>Nb. This answers affirmably questions raised in Erdős and Szemeredi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) and recently, by S. Konjagin (private communication). The method is based on harmonic analysis techniques in the spirit of Chang (Ann. Math. 157 (2003) 939–957) and combinatorics on graphs. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Summary The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd˝ os-Szemeredi (E-S). (see also (El), (T), and (K-T) for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman's structure theorem) (cf. (N-T), (Na3)). We follow the reverse route and prove that if |AA| c � |A| 2 (see Theorem 1). A quantitative version of this phenomenon combined with the Plunnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in (E-S) on the growth in k .I f

In this Note, we present several contributions (or solutions) to problems related to the sizes of sum sets and product sets of integers (or complex numbers), considered in [8]. We also introduce some new methods in this area of research. To cite this article: M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

In this work, we briefly review our recent experimental studies on the observations and waveform dynamics of spontaneous excitations of low and high amplitude singular objects: low amplitude hole filaments coinciding with the wiggling trajectories of topological defects surrounded by acoustic vortices with helical waveforms, and uncertain rogue wave events, in self-excited weakly disordered dust acoustic waves. The changes of waveform topology, caused by kinking, rupturing and reconnection of sequential wave crests surfaces, and the reversed process, are responsible for the chaotic creation, propagation, and annihilation of acoustic vortex pairs with opposite helicities winding around low amplitude hole filaments. The observed rogue wave events are preceded by a higher probability of surrounding defects. Particle focusing by the transverse electric forces from ruptured and tilted wave crests nearby defects are identified as the major cause for rogue wave generation.

A vector bundle has the Bloch-Gieseker property if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle Ω p Pn (p+1) has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of t k in the rational function (1+t) (n/p) (1+3t) (n/p-2) …(1+(p-1)t) (n/2) (1+(p+1)t)/(1+2t) (n/p-1) (1+4t) (n/p-3) …(1+pt) (n/1) (for p even).

Let X⊂ℙN be either a threefold of Calabi–Yau or of general type (embedded with r KX). In this article we give lower and upper bounds, linear on the degree of X and N, for the Euler number of X. As a corollary we obtain the boundedness of the region described by the Chern ratios \(\) of threefolds with ample canonical bundle and a new upper bound for the number of nodes of a complete intersection threefold.

Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a fundamental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families Ng with g [ges ] 2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in Ng is (S, H), where S is a K3 surface and H is a primitive ample divisor on S with H2 = 2g − 2. For a generic (S, H), Pic (S) is generated by H, so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h1(S, TS) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds is irreducible in some sense [R]. A Calabi–Yau threefold is algebraic if and only if it is Kaehler, while every non-algebraic K3 surface is still Kaehler. Inspired by the K3 case, we define Mh,d to be {(X, H)[mid ]H3 = h, c2(X) · H = d}, where H is a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two parameters h, d for algebraic Calabi–Yau threefolds, while there is only one parameter g for algebraic K3 surfaces. (Note that c2(S) = 24 for every K3 surface.) We know that Ng is of dimension 19 for every g and is irreducible but we do not know the dimension of Mh,d and whether or not Mh,d is irreducible. In fact, the dimension of Mh,d = h1(X, TX), where (X, H) ∈ Mh,d. Furthermore, it is well known that χ(X) = 2 (rank of Pic (X) − h1(X, TX)), where χ(X) is the topological Euler characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive [G] and play an important role in the moduli spaces of all Calabi–Yau threefolds. In this paper we give a bound on c3 of Calabi–Yau threefolds with Picard rank 1.