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We prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.

We show that any closed hyperbolic $3$-manifold $M$ has a co-final tower of covers $M_i \to M$ of degrees $n_i$ such that any subgroup of $��_1(M_i)$ generated by $k_i$ elements is free, where $k_i \ge n_i^C$ and $C = C(M) > 0$. Together with this result we show that $\log k_i \geq C_1 sys_1(M_i)$, where $sys_1(M_i)$ denotes the systole of $M_i$, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem $C_1> 0$ is an absolute constant. We also consider a generalization of these results to non-compact finite volume hyperbolic $3$-manifolds., 11 pages; v2: filled a gap in the proof of Theorem 3; v3: includes small corrections to the published version

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\frac{(\Re z)^{3}}{|z|^{4}}+t\cdot \frac{\Re z}{|z|^{2}}, \quad t\in \mathbb{R},\qquad k_\infty(z):=\frac{\Re z}{|z|^{2}}\equiv \Re \frac{1}{z},\quad z\in \mathbb{C}\setminus\{0\}. $$ It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\infty}$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of $T_{k_t}$ with a fixed $t\in (-t_0,0)$, where $t_0>0$ is an absolute constant, implies that $E$ is rectifiable. This is so in spite of the fact that the usual curvature method fails to be applicable in this case. Moreover, as a corollary of our techniques, we provide an alternative and simpler proof of the bi-Lipschitz invariance of the $L^2$-boundedness of the Cauchy transform, which is the key ingredient for the bi-Lipschitz invariance of analytic capacity., In this version, we corrected several inaccuracies and added more corollaries. 1 figure

In a recent breakthrough, Babai (STOC 2016) gave a quasipolynomial time graph isomorphism test. In this work, we give an improved isomorphism test for graphs of small degree: our algorithms runs in time $n^{O((\log d)^{c})}$, where $n$ is the number of vertices of the input graphs, $d$ is the maximum degree of the input graphs, and $c$ is an absolute constant. The best previous isomorphism test for graphs of maximum degree $d$ due to Babai, Kantor and Luks (FOCS 1983) runs in time $n^{O(d/ \log d)}$., 36 pages; second version significantly improves on the results and gives a faster isomorphism test for all graphs of maximum degree d rather than just graphs of maximum degree d and logarithmic diameter; third version adds additional explanations and corrects several typos

Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant $A>0$ such that the equation $R_k(t) = (1+\eta )n$ has at least $An^{0.5394282}$ distinct zeros in $[0,2\pi)$ whenever $\eta$ is real and $|\eta| < 2^{-11}$. In this paper we show that the equation $R_k(t)=(1+\eta)n$ has at least $(1/2-|\eta|-\varepsilon)n/2$ distinct zeros in $[0,2\pi)$ for every $\eta \in (-1/2,1/2)$, $\varepsilon > 0$, and sufficiently large $k \geq k_{\eta,\varepsilon}$., Comment: arXiv admin note: substantial text overlap with arXiv:1708.01189, arXiv:1702.06198

Within the model of social dynamics determined by collective decisions in a stochastic environment (ViSE model), we consider the case of a homogeneous society consisting of classically rational economic agents (or homines economici, or egoists). We present expressions for the optimal majority threshold and the maximum expected capital increment as functions of the parameters of the environment. An estimate of the rate of change of the optimal threshold at zero is given, which is an absolute constant: $(\sqrt{2/\pi}-\sqrt{\pi/2})/2$., Comment: 10 pages, 7 figures

HYPOTHESIS This study investigated the hypothesis that functional outcome remains significantly improved over the preoperative state beyond 15 years of reverse total shoulder arthroplasty (RTSA) for irreparable rotator cuff dysfunction. METHODS Operations were performed on 22 shoulders at a mean age of 68 (range, 54-77) years. The patients could personally be reviewed clinically and radiographically in intervals of 2 to 5 years and with a final follow-up examination at no less than 15 years (mean, 16.1; range, 15-19 years). The RTSA was a primary procedure in 7 procedures, and 15 patients had undergone at least 1 previous nonarthroplasty procedure. RESULTS The mean absolute Constant score (CS) had improved from 23 ± 11 to 58 ± 19 points (P

We study the joint distribution of the solutions to the equation $gh=x$ in $G(\mathbb{F}_p)$ as $p\to\infty$, for any fixed $x\in G(\mathbb{Z})$, where $G=\operatorname{GL}_n$, $\operatorname{SL}_n$, $\operatorname{Sp}_{2n}$ or $\operatorname{SO}_{n}^\pm$. In the special linear case, this answers in particular a question raised by S. Hu and Y. Li, and improves their error terms. Similar results are derived in certain subgroups, and when the entries of $g,h$ lie in fixed intervals. The latter shows for example the existence of $g\in\operatorname{GL}_n(\mathbb{F}_p)$ such that $g,g^{-1}$ have all entries in $[0, c_np^{1-1/(2n^2+2)+\varepsilon}]$ for some absolute constant $c_n>0$. The key for these results is to use Deligne's extension of the Weil conjectures on a sheaf on $G$, along with the stratification theorem of Fouvry, Katz and Laumon, instead of reducing to bounds on classical Kloosterman sums., 16 pages; incorporating the referees' comments

The Corners Theorem states that for any $\alpha > 0$ there exists an $N_0$ such that for any abelian group $G$ with $|G| = N \geq N_0$ and any subset $A \subset G \times G$ with $|A| \ge \alpha N^2$ we can find a corner in $A$ , i.e. there exist $x, y, d \in G$ with $d \neq 0$ such that $(x, y), (x+d, y), (x, y+d) \in A$. Here, we consider a stronger version: given such a group $G$ and subset $A$, for each $d \in G$ we define $S_d = \{(x, y) \in G \times G : (x, y), (x+d, y), (x, y+d) \in A \}$ . So $|S_d|$ is the number of corners of size $d$. Is it true that, provided $N$ is sufficiently large, there must exist some $d \in G \setminus \{0\}$ such that $|S_d|> (\alpha^3 - \epsilon ) N^2$ ? We answer this question in the negative. We do this by relating the problem to a much simpler-looking problem about random variables. Then, using this link, we show that there are sets $A$ with $|S_d| < C\alpha^{3.13} N^2$ for all $d \neq 0$, where $C$ is an absolute constant. We also show that in the special case where $G = \mathbb{F}_2^n$, one can always find a $d$ with $|S_d|> (\alpha^4 - \epsilon ) N^2$., Comment: 17 pages

On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature $\beta$ is small enough, via classical results of Dobrushin and of Holley in the 1970's. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed $\beta>0$, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree $d$, the Ising model has cutoff provided that $\beta, Comment: 26 pages, 2 figures. Companion paper to arXiv:1401.6065

We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets A from R with |AA| \le |A|^{4/3}., 19 pages

Let K be a centered convex body of volume 1 in R n . A direction θ ∈ S n − 1 is called sub-Gaussian for K with constant b > 0 if ‖ 〈 ⋅ , θ 〉 ‖ L ψ 2 ( K ) ⩽ b ‖ 〈 ⋅ , θ 〉 ‖ 2 . We show that if K is isotropic then most directions are sub-Gaussian with a constant which is logarithmic in the dimension. More precisely, for any a > 1 we have ‖ 〈 ⋅ , θ 〉 ‖ L ψ 2 ( K ) ⩽ C ( log ⁡ n ) 3 / 2 max ⁡ { log ⁡ n , a } L K for all θ in a subset Θ a of S n − 1 with σ ( Θ a ) ⩾ 1 − n − a , where C > 0 is an absolute constant.

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most C|VG|. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph Gn,p, which improves upon existing results showing that asymptotically almost surely the cop number of Gn,p is Onlogn provided that pni¾?2+elogn for some e>0. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when d=dni¾?3. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396-421, 2016

We present a deterministic algorithm which, given a graph G with n vertices and an integer 1 0 is an absolute constant. This allows us to tell apart the graphs that do not have m-subsets of high density from the graphs that have sufficiently many m-subsets of high density, even when the probability to hit such a subset at random is exponentially small in m. ACM Classification: F.2.1, G.1.2, G.2.2, I.1.2 AMS Classification: 15A15, 68C25, 68W25, 60C05

We prove that there exists an absolute constant c > 0 such that if an arithmetic progression P modulo a prime number p does not contain zero and has the cardinality less than cp, then it cannot be represented as a product of two subsets of cardinality greater than 1, unless P = − P or P = { − 2 r , r , 4 r } for some residue r modulo p.

The Busemann–Petty problem for an arbitrary measure $$\mu $$ with non-negative even continuous density in $${\mathbb R}^n$$ asks whether origin-symmetric convex bodies in $${\mathbb R}^n$$ with smaller $$(n-1)$$ -dimensional measure $$\mu $$ of all central hyperplane sections necessarily have smaller measure $$\mu .$$ It was shown in Zvavitch (Math Ann 331:867–887, 2005) that the answer to this problem is affirmative for $$n\le 4$$ and negative for $$n\ge 5$$ . In this paper we prove an isomorphic version of this result. Namely, if $$K,M$$ are origin-symmetric convex bodies in $${\mathbb R}^n$$ such that $$\mu (K\cap \xi ^\bot )\le \mu (M\cap \xi ^\bot )$$ for every $$\xi \in {\mathbb S}^{n-1},$$ then $$\mu (K)\le \sqrt{n}\ \mu (M).$$ Here $$\xi ^\bot $$ is the central hyperplane perpendicular to $$\xi .$$ We also study the above question with additional assumptions on the body $$K$$ and present the complex version of the problem. In the special case where the measure $$\mu $$ is convex we show that $$\sqrt{n}$$ can be replaced by $$cL_n,$$ where $$L_n$$ is the maximal isotropic constant. Note that, by a recent result of Klartag, $$L_n \le O(n^{1/4})$$ . Finally we prove a slicing inequality $$\begin{aligned} \mu (K)\le C n^{1/4} \max _{\xi \in {\mathbb S}^{n-1}} \mu (K \cap \xi ^\perp )\ \mathrm{vol}_{n}(K)^{\frac{1}{n}} \end{aligned}$$ for any convex even measure $$\mu $$ and any symmetric convex body $$K$$ in $${\mathbb R}^n,$$ where $$C$$ is an absolute constant. This inequality was recently proved in Koldobsky (Adv Math 254:33–40, 2014) for arbitrary measures with continuous density, but with $$\sqrt{n}$$ in place of $$n^{1/4}.$$

We focus on counting the number of labeled graphs on $n$ vertices and treewidth at most $k$ (or equivalently, the number of labeled partial $k$-trees), which we denote by $T_{n,k}$. So far, only the particular cases $T_{n,1}$ and $T_{n,2}$ had been studied. We show that $$ \left(c \cdot \frac{k\cdot 2^k \cdot n}{\log k} \right)^n \cdot 2^{-\frac{k(k+3)}{2}} \cdot k^{-2k-2}\ \leq\ T_{n,k}\ \leq\ \left(k \cdot 2^k \cdot n\right)^n \cdot 2^{-\frac{k(k+1)}{2}} \cdot k^{-k}, $$ for $k > 1$ and some explicit absolute constant $c > 0$. The upper bound is an immediate consequence of the well-known number of labeled $k$-trees, while the lower bound is obtained from an explicit algorithmic construction. It follows from this construction that both bounds also apply to graphs of pathwidth and proper-pathwidth at most $k$., 12 pages, 3 figures

It is shown that the maximum of $\vert \zeta(1/2+it)\vert $ on the interval $T^{1/2}\le t\le T$ is at least $\exp ((1/\sqrt{2}+o(1))\sqrt{\log T\log\log\log T/\log\log T})$ . Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant $A$ in the inequality ¶ \[\sup_{1\le n_{1}\lt \cdots\lt n_{N}}\sum_{k,{\ell}=1}^{N}\frac{\operatorname{gcd}(n_{k},n_{\ell})}{\sqrt{n_{k}n_{\ell}}}\ll N\exp (A\sqrt{\frac{\log N\log\log\logN}{\log\log N}}),\] established in a recent paper of ours, cannot be taken smaller than $1$ .

We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)\leq R_k(P_n)\leq R_k(C_n)\leq kn+o(n)$. The upper bound was recently improved by S\'ark\"ozy who showed that $R_k(C_n)\leq\left(k-\frac{k}{16k^3+1}\right)n+o(n)$. Here we show $R_k(C_n) \leq (k-\frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant., Comment: 12 pages, minor changes to exposition

We prove that there is an absolute constant $ C$ such that for every $ n \geq 2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n $ with at most $ N $ facets that satisfies $$\Delta_{v}(D_n,P_{n,N}):=\text{vol}_n\left(D_n \Delta P_{n,N}\right)\leq Cn^{-2/(n-1}\text{vol}_n\left(D_n\right)$$ and $$ \Delta_{s}(D_n,P_{n,N}):=\text{vol}_{n-1}\left(\partial\left(D_n\cup P_{n,N}\right)\right) - \text{vol}_{n-1}\left(\partial\left(D_n\cap P_{n,N}\right)\right) \leq 4CN^{-\frac{2}{n-1}} \text{vol}_{n-1}\left(\partial D_n\right), $$ where $ D_n $ is the $ n$-dimensional Euclidean unit ball. This result closes gaps from several papers of Hoehner, Ludwig, Schutt and Werner. The upper bounds are optimal up to absolute constants. This result shows that a polytope with an exponential number of facets (in the dimension) can approximate the $ n$-dimensional Euclidean ball with respect to the aforementioned distances.

The projection function $$P_K$$PK of an origin-symmetric convex body K in $${{\mathbb {R}}}^n$$Rn is defined by $$P_K(\xi )=|K\vert {\xi ^\bot }|,\ \xi \in S^{n-1},$$PK(ź)=|K|źź|,źźSn-1, where $$K\vert {\xi ^\bot }$$K|źź is the projection of K to the central hyperplane $$\xi ^\bot $$źź perpendicular to $$\xi $$ź, and |K| stands for volume of proper dimension. We prove several stability and separation results for the projection function. For example, if D is a projection body in $${{\mathbb {R}}}^n$$Rn which is in isotropic position up to a dilation, and K is any origin-symmetric convex body in $${{\mathbb {R}}}^n$$Rn such that that there exists $$\xi \in S^{n-1}$$źźSn-1 with $$P_K(\xi )>P_D(\xi ),$$PK(ź)>PD(ź), then $$\begin{aligned} \max _{\xi \in S^{n-1}} (P_K(\xi )-P_D(\xi )) \ge \frac{c}{\log ^2n} \big (|K|^{\frac{n-1}{n}} -|D|^{\frac{n-1}{n}}\big ), \end{aligned}$$maxźźSn-1(PK(ź)-PD(ź))źclog2n(|K|n-1n-|D|n-1n),where c is an absolute constant. As a consequence, we prove a hyperplane inequality $$\begin{aligned} S(D) \le \ C \log ^2n \max _{\xi \in S^{n-1}} S(D\vert \xi ^\bot )\ |D|^{\frac{1}{n}}, \end{aligned}$$S(D)≤Clog2nmaxźźSn-1S(D|źź)|D|1n,where D is a projection body in isotropic position, up to a dilation, S(D) is the surface area of $$D,S(D\vert \xi ^\bot )$$D,S(D|źź) is the surface area of the body $$D\vert \xi ^\bot $$D|źź in $${{\mathbb {R}}}^{n-1},$$Rn-1, and C is an absolute constant. The proofs are based on the Fourier analytic approach to projections developed in [12].

In this work we consider the first order linear difference equation −yt+1+(1+rt)yt=et, t=0,1,2,…, where e ∈ lp, 1 ≤ p ≤ ∞, rt ≥ 0 and ∏j=0T−1(1+rt+j)

We provide a new quantitative version of Helly's theorem: there exists an absolute constant $$\alpha >1$$ź>1 with the following property. If $$\{P_i: i\in I\}$${Pi:iźI} is a finite family of convex bodies in $${\mathbb {R}}^n$$Rn with $${\mathrm{int}} (\bigcap _{i\in I}P_i )\ne \emptyset $$int(źiźIPi)źź, then there exist $$z\in {\mathbb {R}}^n$$zźRn, $$s\leqslant \alpha n$$sźźn and $$i_1,\ldots i_s\in I$$i1,źisźI such that $$\begin{aligned} z+P_{i_1}\cap \cdots \cap P_{i_s}\subseteq cn^{3/2}\Big (z+\bigcap _{i\in I}P_i\Big ), \end{aligned}$$z+Pi1źźźPis⊆cn3/2(z+źiźIPi),where $$c>0$$c>0 is an absolute constant. This directly gives a version of the "quantitative" diameter theorem of Barany, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound $$O(n^{3/2})$$O(n3/2) can be improved to $$O(\sqrt{n})$$O(n).

Siegel’s Lemma is concerned with finding a “small” nontrivial integer solution of a large system of homogeneous linear equations with integer coefficients, where the number of variables substantially exceeds the number of equations (for example, n equations and N variables with N ≥ 2n), and “small” means small in the maximum norm. Siegel’s Lemma is a clever application of the Pigeonhole Principle, and it is a pure existence argument. The basically combinatorial Siegel’s Lemma is a key tool in transcendental number theory and diophantine approximation. David Masser (a leading expert in transcendental number theory) asked the question whether or not the Siegel’s Lemma is best possible. Here we prove that the so-called “Third Version of Siegel’s Lemma” is best possible apart from an absolute constant factor. In other words, we show that no other argument can beat the Pigeonhole Principle proof of Siegel’s Lemma (apart from an absolute constant factor). To prove this, we combine a concentration inequality (i.e., Fourier analysis) with combinatorics.

In the central limit theorem for sums of independent random variables with finite absolute moments of order 2 + δ , δ ∈ (0, 1], nonuniform analogues of the Berry–Esseen-type inequalities with an improved structure are proposed. An analytical representation and an algorithm of evaluation of the absolute constants in the proposed estimates is given, in particular, for the absolute constant C ( δ ) in Nagaev–Bikelis inequality with δ = 1 upper bounds C(1)⩽17.36 C ( 1 ) ⩽ 17.36 in the i.i.d. case and C(1)⩽21.82 C ( 1 ) ⩽ 21.82 in the general situation are obtained. Moreover, it is proposed to replace the absolute constants C ( δ ) by nonincreasing functions C ( δ , x ) of the argument x of distribution functions under consideration, for which improved upper bounds are obtained yielding an asymptotic estimate lim |x|→∞ C(δ,x)⩽1+e lim | x | → ∞ C ( δ , x ) ⩽ 1 + e 3.72 for every δ ∈ (0, 1]. Similar result was presented by Nikulin [ 41 ] in the i.i.d. case with δ = 1. As corollary, upper bounds for the Kolmogorov asymptotically exact functions are given.

We derived an asymptotic formula for the number of pairs of integers which are mutually squares. Earlier results dealt with pairs of integers subject to the restriction that they are both odd, co-prime and squrefree. Here we remove all these restrictions and prove (similar to the best known one with restrictions) that the number of such pair of integers upto a large real X is asymptotic to \(\frac{{c{X^2}}}{{\log X}}\) with an absolute constant c which we give explicitly. Our error term is also compatible to the best known one.

We prove that the uniform probability measure μ on every ( n − k ) -dimensional projection of the n-dimensional unit cube verifies the variance conjecture with an absolute constant C Var μ | x | 2 ≤ C sup θ ∈ S n − 1 ⁡ E μ 〈 x , θ 〉 2 E μ | x | 2 , provided that 1 ≤ k ≤ n . We also prove that if 1 ≤ k ≤ n 2 3 ( log ⁡ n ) − 1 3 , the conjecture is true for the family of uniform probabilities on its projections on random ( n − k ) -dimensional subspaces.

Due to the effort of a number of authors, the value c u of the absolute constant factor in the uniform Berry–Esseen (BE) bound for sums of independent random variables has been gradually reduced to 0.4748 in the iid case and 0.5600 in the general case; both these values were recently obtained by Shevtsova. On the other hand, Esseen had shown that c u cannot be less than 0.4097. Thus, the gap factor between the best known upper and lower bounds on (the least possible value of) c u is now rather close to 1. The situation is quite different for the absolute constant factor c nu in the corresponding nonuniform BE bound. Namely, the best correctly established upper bound on c nu in the iid case is over 17 times the corresponding best known lower bound, and this gap factor is greater than 21 in the general case. In the present paper, improvements to the prevailing method (going back to S. Nagaev) of obtaining nonuniform BE bounds are suggested. Moreover, two new methods are presented, of a rather purely Fourier kind, based on two families of smoothing inequalities, which work better in the tail zones. As illustrations, two corresponding quick proofs of Nagaev’s nonuniform BE bound are given. Some further refinements in the application of the methods are shown as well.

We prove that there exists an absolute constant C so that $$\displaystyle{ \mu (K)\ \leq \ C\sqrt{p}\max _{\xi \in S^{n-1}}\mu (K \cap \xi ^{\perp })\ \vert K\vert ^{1/n} }$$ for any p > 2, any \(n \in \mathbb{N},\) any convex body K that is the unit ball of an n-dimensional subspace of L p , and any measure μ with non-negative even continuous density in \(\mathbb{R}^{n}.\) Here ξ ⊥ is the central hyperplane perpendicular to a unit vector ξ ∈ Sn−1, and | K | stands for volume.

Suppose that an infinite set $${\fancyscript{A}}$$ occupies at most $${\frac{1}{2}(p+1)}$$ residue classes modulo p, for every sufficiently large prime p. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $${\fancyscript{A}}$$ that are at most X is $${O(X^{1/2})}$$ , and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. We prove a variety of results and formulate various conjectures in connection with this problem, including several improvements of the large sieve bound when the residue classes occupied by $${\fancyscript{A}}$$ have some additive structure. Unfortunately we cannot solve the problem itself.

The interrelation between signs and permutations in maximum inequalities is studied in this paper. The relationship is based on a lemma that reduces a rearrangement problem to a problem of choosing signs. It helps simplify proofs and find new facts and general settings. The following inequality is one of the main results of the paper: Let $x_1,\ldots , x_n$, $\sum_{k=1}^n x_k =0,$ be a collection of elements of a normed space $X$. Then for any collection of signs $\theta = (\theta_1 , \ldots , \theta_n)$ and any $t>0$, $ {\bf P} \{\pi : \max_{1\le k \le n}\Vert\sum_1^k x_{\pi (i)}\Vert > t\} \le C{\bf P}\{\pi : \max_{1\le k \le n}\Vert \sum_1^k x_{\pi (i)}\theta_i\Vert> \frac {t}{C}\}, $ where $\pi\in \Pi_n ,$ $\Pi_n$ is the group of all permutations of $\{1,\ldots ,n\}$, $P$ is the uniform distribution on it, and $C$ is an absolute constant. The inequality is unimprovable (the inverse inequality also holds with some other constant); it generalizes well-known results due to Garsia; Maurey, and Pisier; Kas...

We show that there is an absolute constant $c \leq 156$ such that in every finite family ${\cal F}$ of pseudodiscs in the plane one can find a member $D \in {\cal F}$ such that among all of the pseudodiscs in ${\cal F}$ intersecting $D$ there are at most $c$ pairwise disjoint sets.

Background: Due to the rotator cuff retear after being surgically repaired, some strategies have been developed. The authors verified that the possibility of polyetheretherketone (PEEK) vented anchors promoted a better clinical and healing process than PEEK solid anchors.Methods: A prospective and randomized study was designed with 38 patients treated with PEEK anchors, 18 of whom with vented anchors and 20 with solid ones. Demographic, clinical and radiologic data were collected before and during surgery (time 0) and at 12 months of follow-up.Results: In the final follow-up (12 months), there was no difference in the visual analogic scale (VAS) scale between groups (1.7 points vs 1.9 points;p = 0.731), neither in the DASH score (34.2 points vs 23.9 points;p = 0.268), nor in absolute Constant score (76.9 points vs 77.3 points;p = 0.910). In MRI, 10 patients had their cuff tear healed in the vented group and 15 in the solid group (p = 0.173).Conclusion: The new designed vented anchors do not add any advantage when compared to solids ones, at least within the first year after surgery.

Let psi: R^n --> R^k be a map defined by k positive definite quadratic forms on R^n. We prove that the relative entropy (Kullback-Leibler) distance from the convex hull of the image of psi to the image of psi is bounded above by an absolute constant. More precisely, we prove that for every point a=(a_1, ..., a_k) in the convex hull of the image of psi such that a_1 + ... +a_k =1 there is a point b=(b_1, ..., b_k) in the image of psi such that b_1 + ... + b_k =1 and such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k) < 4.8. Similarly, we prove that for any integer m one can choose a convex combination b of at most m points from the image of psi such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k) < 15/sqrt{m}., 16 pages

We prove that for any symmetric n -dimensional normed space E and any e > 0 , E contains a ( 1 + e ) -Euclidean subspace of dimension at least c ln n / ln ( 1 / e ) , where c is an absolute constant. The proof is based on a concentration property of order statistics of random vectors with i.i.d. coordinates.

For various k-uniform hypergraphs F, we give tight lower bounds on the number of copies of F in a k-uniform hypergraph with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of various authors who proved that there is one copy of F. A sample result is the following: Furedi-Simonovits [11] and independently Keevash-Sudakov [16] settled an old conjecture of Sos [29] by proving that the maximum number of triples in an n vertex triple system (for n sufficiently large) that contains no copy of the Fano plane is p(n)=( 2 ?n/2? )?n/2?+( 2 ?n/2? ?n/2?). We prove that there is an absolute constant c such that if n is sufficiently large and 1 ≤ q ≤ cn 2, then every n vertex triple system with p(n)+q edges contains at least $6q\left( {\left( {_4^{\left\lfloor {n/2} \right\rfloor } } \right) + \left( {\left\lceil {n/2} \right\rceil - 3} \right)\left( {_3^{\left\lfloor {n/2} \right\rfloor } } \right)} \right)$ copies of the Fano plane. This is sharp for q≤n/2---2. Our proofs use the recently proved hypergraph removal lemma and stability results for the corresponding Turan problem.

Let P be a set of n blue points in the plane, not all on a line. Let R be a set of m red points such that P ∩ R = ∅ and every line determined by P contains a point from R. We provide an answer to an old problem by Grunbaum and Motzkin [9] and independently by Erdős and Purdy [6] who asked how large must m be in terms of n in such a case? More specifically, both [9] and [6] were looking for the best absolute constant c such that m ≥ cn. We provide an answer to this problem and show that m ≥ (n−1)/3.

Let A be a finite set of integers and FA(x) = ∑a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that ∥FA∥1 ≥ c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L1-norm of exponential sums of sets in the d-dimensional grid d. We show that ∥FA∥1 is considerably larger than log|A| when A ⊂ d has multidimensional structure. We furthermore prove similar lower bounds for sets in , which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

In this paper we prove the following theorem for simultaneous approximation in L spaces for p r,there exists a polynomial p of degree n, such that for all j = 0, . . . , mwe have|f z p z | c ωA∗ .f ; 0 , ∀z ∈ ∂ ,and p zC # f zC , l # 1,... , q,Where C is independent of n and z.HereωA∗ g; δ : # supG∈ : HEA -Ig; : ∩ B z; δ LM,B(N;δ # %ξ ∈ "; |ξ N| δ;, E g;M :# inf %‖g p‖Q; P complex polynomial of degree m;.Theorem 1.1was proved by (N.N.Vorob’ev) in [6] for the so-called domains of type1.1in the complex plane (including the unit disk) and Theorem1.2 was proved by (V.V. Andrievskii, I. Pritsker and R. Varga)in [1] for general continuain the complex plane (including the unit disk).In our paper , we will re-state them here for the particular case of unit disk only. Unfortunately, the constants appearing in these estimates, are claimed in the corresponding papers, as independent of n and zonly, without to be mentioned the independence of ftoo. Because of complicated technical details, it seems to be very difficult to deduce from the proofs of Theorems 1.1and 1.2that possibly these constants are independent of f too. For this reason, in the case of unit disk, by our main result Theorem 3.1, in the following section we improve these theorems to the spaces L ,for p R 1 with new simple proof which clearly shows that the constant is independent of n, z and f too. 2. The Auxiliary results To prove our theorem we need the following definitions and results for f ∈ A , σ ,T f z # TU-∑ TX XY T f z ,where TX f z # ∑ Z [ \ ! X Y\ z Lemma 2.1: T^X f z # TX f ^ z . Proof: TX f z # ∑ Z [ \ ! X Y\ z ,k# 0,... ,mTX f z # ∑ j Z [ \ ! X Y\ z # ∑ j Z [`a \ b ! X Y\ z # ∑ IZ [ Lb \ ! X Y\ z # TX f ^)(z)∎ Lemma 2.2: σd , X f z # σd X, If X L z Simultaneous approximation by complex polynomials 1805 Proof: σ ,T f z # TU-∑ TX XY T f z wheren # 2n, h # n m ,k=0,...,m,σd , f z # U-∑ TX d XYd U f z bylemma2.1weget σd , ^ f z # 1 n m> 1 f TX d XYd U f ^ z # σd -, f ^ z σ^^d , f z # 1 n m> 1 f TX d d XYd U f ^^ z # σd d, f ^^ z σ^^^d , f z # 1 n m> 1 f TX g d XYd U f ^^^ z # σd g, f ^^^ z ⋮ σd , X f z # 1 n m> 1 f TX X d XYd U fX z # σd X, fX z ∎ Lemma 2.3 [5]:‖f σ , f ‖i A∑ jk`lm[ Z n U UY\ where: E f i # inf %‖f p‖i, p is polynomail of degree } ,A is an absolute constant independent of f, n and m. Lemma 2.4[3]: supp∈qr,st|p x | vw s r‖p ‖x qr,st ,0 R y R ∞ Proposition 2.5: ∥ f σ ,T f ∥ c p ∑ jk`lm[ Z w U UY\ where c p is an absolute constant independent of f, n and m. Proof: It is sufficient to show E U f i c p E U f Assume E f #∥ f Q ∥~ , 0 R  R ∞ , €‚ƒ „ …†ƒ ∈ ‡ €ˆ‰  Š ‹Œ 2C # n AndPd be apolynomial of best approximation of f.Then we may writef p # ∑ pdk[ i Ypdk[`a .Thus S.N.Bernstein inequality [2] yieldsE f i #∥ ∑ pd Y\ pd ∥iThen by lemma 2.4we obtain E f i c p ∥ fpd pd Y\ ∥ c p ∥ f Q ∥~ ......... 1 1806 Eman Samir Bhaya and Asmaa Raheem Hadi # c p E f Hence∥ f σ , f ∥ ∥ f σ , f ∥i A∑ jk`lm[ Z n U UY\ Then from(1)we get ∥ f σ , f ∥ c p ∑ jk`lm[ Z w U UY\ ∎ Lemma 2.6: Ifn R > 1,then E Uf E f Proof: E Uf # inf kma∈∏kma ∥ f p U∥ , where ∏ ∏ Uinf k∈∏k ∥ f p ∥ # E f ∎ Lemma 2.7:∥ f σd X, f ∥ c p E U X f where c p is an absolute constant independent of f, n and m. Proof: UsingProposition 2.5 to obtain ∥ f σd X, f ∥ c p f E U XU f n m > j > 1 d X Y\ c p E U X f f 1 n m> j > 1 d X Y\ c p E U X f 2n k > 1 n m> 1 byn msothat∥ f σd X, f ∥ c p E U X f d UUc p 2m > 1 E U X f ∎ Lemma 2.8[4]: E f i c p n E f i where c p is a constant depending only on p. Proposition 2.9: E f c p n E f where c p is a constant depending only on p. Proof: Using lemma2.8 we have  f c p E f i c p n E f i Using the same lines used for the proof of proposition 2.5we can get E f c p E f i c p n E f i c p n E f ∎ 3. The main result Theorem 3.1:∥ f X P X ∥ c p n UXE f Simultaneous approximation by complex polynomials 1807 where c p is a constant depending only on p. Proof: ∥ f X P X ∥ ∥ f X σd , X f ∥ >∥ σd , X f p X ∥ Usinglemma2.2weget∥ f X P X ∥ #∥ f X σd X, f X ∥ >∥ σd , X f p X ∥ Usinglemma2.7weobtain ∥ f X P X ∥ c p E U X f X >∥ σd , f p X ∥ by Bernstein’s inequality we have∥ f X P X ∥ c p E U X f X > 2n X ∥ σd , f p ∥ c p E U X f X > C p nXq∥ σd , f f ∥ >∥ f P ∥ c p E U X f X > C p nXqE U f > E f t c p E U X f X > C p nXE f byProposition2.9weget∥ f X P X ∥ C p E U X f X > C p nXn E f by Proposition 2. 9we obtain ,for n > m k ∥ f X P X ∥ C p n > m k UXE f > C p n UXE f C p n UXE f ∎