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Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erd\H{o}s matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots, F_m\subseteq {[n]\choose k}$ has size more than $\max\{\binom{n}{k} - \binom{n-m+1}{k}, \binom{km-1}{k}\}$, then there exist pairwise disjoint subsets $e_1,\dots, e_m$ such that $e_i\in F_i$ for all $i\in [m]$. We prove that there exists an absolute constant $n_0$ such that this rainbow version holds for $k=3$ and $n\geq n_0$. We convert this rainbow matching problem to a matching problem on a special hypergraph $H$. We then combine several existing techniques on matchings in uniform hypergraphs: find an absorbing matching $M$ in $H$; use a randomization process of Alon et al. to find an almost regular subgraph of $H-V(M)$; and find an almost perfect matching in $H-V(M)$. To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of {\L}uczak and Mieczkowska and might be of independent interest., Comment: added two references [2,7], accepted for publication in SCIENCE CHINA Mathematics

Let $$G$$ be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant $$c$$ such that whenever $$S$$ is a non-trivial normal subset in $$G$$ then $$S^{k} = G$$ for any integer $$k$$ at least $$c \cdot (\log|G|/\log|S|)$$ . This result is generalized by showing that there exists an absolute constant $$c$$ such that whenever $$S_{1}$$ , $$\ldots , $$ $$S_{k}$$ are normal subsets in $$G$$ with $$\prod_{i=1}^{k} |S_{i}| \geq {|G|}^{c}$$ then $$S_{1} \cdots S_{k} = G$$ .

An example is presented of a subset $A$ of $\mathbb Z_N$ of density $\alpha$ such that the largest non-trivial Fourier coefficient of the characteristic function of $A$ is very small, but the probability that a random arithmetic progression (mod $N$) of length 4 lies in $A$ is significantly smaller than $\alpha^4$., Comment: This preprint was written well over a decade ago and has been known about by experts but not made publicly available. I apologize for this and am now remedying the situation. Revised version updates the notation, corrects minor errors (pointed out by referee), and adds one or two references

There is an absolute constant D0 > 0 such that if f(x) is an integer polynomial, then there is an integer λ with |λ| ≤ D0 such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. Furthermore, if deg f ≤ 25, then there is a λ with λ ∈ {−2, −1, 0, 1, 2, 3} such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pal Turan.

We prove that there is an absolute constant A > 0 such that $$\begin{array}{l} \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}|P'(x)| \ge A\sqrt {n \cdot } \begin{array}{*{20}{c}} {\max } \\ { - 1 \le x \le 1} \\ \end{array}\,|P(x)| \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{array}$$ for an arbitrary algebraic polynomial P of degree n whose zeros lie in the half-disk $$\left\{ {z:|z| \le 1,{\mathop{\rm Im}\nolimits} z \ge 0} \right\}$$.

A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors $$p_{1},\ldots, p_{t}$$ of n and positive integers $$\alpha_{1}, \ldots , \alpha_{t}$$ such that $$n = p_{1}^{\alpha_{1}}+ \cdots + p_{t}^{\alpha_{t}}$$. Erdős and Hegyvari [2] proved that, for any prime p, there exists a weakly prime-additive number which is divisible by p. Recently, Fang and Chen [3] proved that for any given positive integer m, there are infinitely many weakly prime-additive numbers which are divisible bym with t = 3 if and only if $$8 \nmid m$$. In this paper, we prove that for any given positive integer m, the number of weakly prime-additive numbers which are divisible by m and less than x is larger than $${\rm exp}(c({\rm log log} x)^{2}/ {\rm log log log} x)$$ for all sufficiently large x, where c is a positive absolute constant. The constant c depends on the result on the least prime number in an arithmetic progression.

Green proved an arithmetic analogue of Szemer\'edi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of $\mathbb{F}_p^n$ with $n$ sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive $\epsilon$. For a fixed odd prime $p$, we prove that the required dimension grows as an exponential tower of $p$'s of height $\Theta(\log(1/\epsilon))$. This improves both the lower and upper bound, and is the first example of a result where a tower-type bound coming from applying a regularity lemma is shown to be necessary., Comment: 18 pages

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0, Comment: Version 2: typos corrected and a new section on natural density added

We show that there exists an absolute constant $c > 0$, such that, for any finite set $A$ of quaternions, \[ \max\{|A+A, |AA| \} \gtrsim |A|^{4/3 + c}. \] This generalizes a sum-product bound for real numbers proved by Konyagin and Shkredov., Appeared in SIAM J. Discrete Math. This version corrects some errors from the previous version, and clarifies the analysis

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there is an integer n > 1 such that the ball of radius n in some Cayley graph of G has cardinality bounded by n^(c log log n), then G is virtually (log n)-step nilpotent., 15 pages, 1 figure. Accepted manuscript (to appear in Math. Ann.)

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq C\frac {\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon >0$ and $n> n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac {\log n}{\log \log n}$. This proves the conjecture of Erd\H{o}s, Faudree, Rousseau and Schelp for large $\ell $, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd\H{o}s, Faudree, Rousseau and Schelp., Comment: 19 pages

We show the existence of an absolute constant $\alpha>0$ such that, for every $k \geq 3$, $G:=\mathop{\mathrm{Sym}}(k)$, and for every $H \leqslant G$ of index at least $3$, one has $|H/[H,H]| \leq |G:H|^{\alpha/ \log \log |G:H|}$. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups., Comment: 6 pages

For an integer $m \geq 3$, does there exist an absolute constant $K(m)$ such that every polynomial with $m$ non-zero coefficients has an irreducible factor with at most $K(m)$ coefficients? A previous result in the literature establishes $K(3) \geq 9$, which is here improved to $K(3) \geq 12$. Improvements on known bounds are also given for $m=4,5,6$, and for $K(m)$, when $m \geq 7$.

O’Hara introduced several functionals as knot energies. One of them is the Mobius energy. We know its Mobius invariance from Doyle-Schramm’s cosine formula. It is also known that the Mobius energy was decomposed into three components keeping the Mobius invariance. The first component of decomposition represents the extent of bending of the curves or knots, while the second one indicates the extent of twisting. The third one is an absolute constant. In this paper, we show a similar decomposition for generalized O’Hara energies. We also extend the cosine formula for the Mobius energy to generalized O’Hara energies. It gives us a condition for which the right circle minimizes the energy under the length-constraint. Furthermore, it shows us how far the energy is from the Mobius invariant property. Using decomposition, the first and second variation formulae are derived.

Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\] where $\delta >0$ is an absolute constant only depending on $d$. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu., Comment: 14 pages

Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |\{|R_jf|>\lambda\}|\leq C\left(\frac{1}{\lambda}\|f\|_{L^1(\mathbb{R}^n)}+\sup_{\nu} |\{|R_j\nu|>\lambda\}|\right) \end{align*} for any $\lambda>0$ and $f \in L^1(\mathbb{R}^n)$, where the above supremum is taken over measures of the form $\nu=\sum_{k=1}^Na_k\delta_{c_k}$ for $N \in \mathbb{N}$, $c_k \in \mathbb{R}^n$, and $a_k \in \mathbb{R}^+$ with $\sum_{k=1}^N a_k \leq 16\|f\|_{L^1(\mathbb{R}^n)}$. This shows that to establish dimensional estimates for the weak-type $(1,1)$ inequality for the Riesz tranforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calder\'on-Zygmund operators., Comment: 17 pages

Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tilde{\Lambda}(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$ |K|^{n-1}\leqslant \tilde{\Lambda}(K)\prod_{i=1}^n|K\cap w_i^\perp|. $$ We prove that $\tilde{\Lambda}(K)\leqslant(CL_K)^n$ for some absolute $C>1$ and that this estimate in terms of $L_K$, the isotropic constant of $K$, is asymptotically sharp in the sense that there exists another absolute constant $c>1$ and a convex body $K$ such that $(cL_K)^n\leqslant\tilde{\Lambda}(K)\leqslant(CL_K)^n$. We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.

We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials., Inaccuracies corrected following the referee's suggestions

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.

We show that there exists an absolute constant $C>0$ such that any family $\mathcal{F}\subset \{0,1\}^n$ of size at least $Cn^3$ has dual VC-dimension at least 3. Equivalently, every family of size at least $Cn^3$ contains three sets such that all eight regions of their Venn diagram are non-empty. This improves upon the $Cn^{3.75}$ bound of Gupta, Lee and Li and is sharp up to the value of the constant., 11 pages

We give quantitative generalized Voronovskaja’s formulae for Bernstein polynomials which are simple to compute. To achieve this, we obtain explicit and accurate upper estimates for the even central moments of Bernstein polynomials. As a consequence, we improve the value of the absolute constant in Videnskiĭ type inequalities.

We show that there is an absolute constant $$c > 1/2$$ such that the Mahler measure of the Fekete polynomials $$f_p$$ of the form $$\begin{aligned} f_p(z) := \sum _{k=1}^{p-1}{\left( \frac{k}{p} \right) z^k} \end{aligned}$$ (where the coefficients are the usual Legendre symbols) is at least $$c\sqrt{p}$$ for all sufficiently large primes p. This improves the lower bound $$\left( \frac{1}{2} - \varepsilon \right) \sqrt{p}$$ known before for the Mahler measure of the Fekete polynomials $$f_p$$ for all sufficiently large primes $$p \ge c_{\varepsilon }$$ . Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

Define to be the largest cardinality of a set that does not contain four elements in arithmetic progression. In 1998, Gowers proved that for some absolute constant . In 2005, the authors improved this to In this paper we further improve this to which appears to be the limit of our methods.

Partly on the basis of heuristic arguments from physics, it has been suggested that the performance of certain types of algorithms on random $k$-SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying assignments. But, beyond intuition, there has been scant rigorous evidence that “practical” algorithms are affected by these phase transitions. In this paper we prove that Walksat, a popular randomized satisfiability algorithm, fails on random $k$-SAT formulas not very far above clause/variable density, where the set of satisfying assignments shatters into tiny, well-separated clusters. Specifically, we prove that Walksat is ineffective with high probability (w.h.p.) if $m/n>c2^k\ln^2k/k$, where $m$ is the number of clauses, $n$ is the number of variables, and $c>0$ is an absolute constant. By comparison, Walksat is known to find satisfying assignments in linear time w.h.p. if $m/n 0$ [A. Coja-Oghlan and A. Frieze, SIAM J. Comput., 43 (2...

Let \({\mathcal {L}}\) be the set of integers n which can be written as $$ n=p_1^3 + p_2^3+p_3^3+p_4^3. $$ Using the circle method and sieves, we prove that \({\sum_{N 0}\) is an absolute constant.

We prove that there exists an absolute constant $${\alpha > 1}$$ with the following property: if K is a convex body in $${{\mathbb R}^n}$$ whose center of mass is at the origin, then a random subset $${X\subset K}$$ of cardinality $${{\rm card}(X)=\lceil\alpha n\rceil }$$ satisfies with probability greater than $${1-e^{-c_1n}}$$ $$K\subseteq c_2n\, {\rm conv}(X),$$ where $${c_1, c_2 > 0}$$ are absolute constants. As an application we show that the vertex index of any convex body K in $${{\mathbb R}^n}$$ is bounded by $${c_3n^2}$$ , where $${c_3 > 0}$$ is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.

Some of known inequalities for the uniform distance between distributions of sequential sums of independent identically distributed random variables are considered. In the case where the distribution F has 0 as the q-quantile, an upper bound for the absolute constant in the inequality is given.

In this paper, we prove the following two results. Let $d$ be a natural number and $q,s$ be co-prime integers such that $1 < qs$. Then there exists a constant $\delta > 0$ depending only on $q,s$ and $d$ such that for any finite subset $A$ of $\mathbb{R}^d$ that is not contained in a translate of a hyperplane, we have $$ |q\cdot A + s\cdot A| \geq (|q| +|s|+ 2d-2)|A| - O_{q,s,d}(|A|^{1-\delta}) . $$ The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let $\mathscr{L} \in \textrm{GL}_{2}( \mathbb{R})$ be a linear transformation such that $\mathscr{L}$ does not have any invariant one-dimensional subspace of $\mathbb{R}^2$. Then for all finite subsets $A$ of $\mathbb{R}^2$, we have $$ |A + \mathscr{L}(A)| \geq 4|A| - O(|A|^{1-\delta}), $$ for some absolute constant $\delta > 0$. The main term in this result is sharp as well., Comment: 17 pages, minor correction in statement of Theorem 1.1

A subset {g1, …, gd} of a finite group G is said to invariably generate G if the set $${\rm{\{ }}g_1^{{x_1}}{\rm{,}} \ldots {\rm{,}}g_d^{{x_d}}{\rm{\}}}$$ generates G for every choice of xi ∈ G. The Chebotarev invariant C(G) of G is the expected value of the random variable n that is minimal subject to the requirement that n randomly chosen elements of G invariably generate G. The authors recently showed that for each ϵ > 0, there exists a constant cϵ such that $$C(G) \le (1 + \epsilon)\sqrt {{\rm{|}}G{\rm{|}}} + {c_{\epsilon}}$$. This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each α > 0 there exists an absolute constant δα such that if G is a finite group and $$C(G) > \alpha \sqrt {{\rm{|}}G{\rm{|}}}$$, then G has a section X/Y such that $${\rm{|}}X/Y{\rm{|}} \ge {\delta _\alpha}\sqrt {{\rm{|}}G{\rm{|}}} $$, and $$X/Y \cong {\mathbb{F}_q}\rtimes H$$ for some prime power q, with $$H \le \mathbb{F}_q^\times $$.

The well-known extremal problem on hypergraph colorings is studied. We investigate whether it is possible to color a hypergraph with a fixed number of colors equitably, i.e., so that, on the one hand, no edge is monochromatic and, on the other hand, the cardinalities of the color classes are almost the same. It is proved that if H = (V,E) is a simple hypergraph in which the least cardinality of an edge equals k, |V| = n, r|n, and $$\sum\limits_{e \in E} {{r^{1 - \left| e \right|}}} \leqslant c\sqrt k ,$$ where c > 0 is an absolute constant, then there exists an equitable r-coloring of H.

A classical inequality of Rogers and Shephard states that if K is a centered convex body of volume 1 in Rn then 1 6 g(K, k;F ) := ( volk(PF (K)) voln−k(K ∩ F⊥) )1/k 6 (n k )1/k 6 cn k for every F ∈ Gn,k, where c > 0 is an absolute constant. We show that if K is origin symmetric and isotropic then, for every 1 6 k 6 n − 1, a random F ∈ Gn,k satisfies c1L −1 K √ n/k 6 g(K, k;F ) 6 c2 √ n/k (logn)LK with probability greater than 1 − e−k, where LK is the isotropic constant of K and c1, c2 > 0 are absolute constants.

Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the sums of the form $\sum_{x \in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same colour. In 1989, Erd\H{o}s and Spencer showed that $F(k) \ge 2^{ck^2/ \log k}$, where $c >0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \ge 2^{2^{k-1}/k}$ for all $k\in \mathbb{N}$., Comment: 5 pages, Bulletin of the LMS

A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $\mathbb{R}^{2}$ of length one. The classical "worm problem" of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family has always area at least $c$ for some small absolute constant $c > 0$. We strengthen Marstrand's result by showing that for $p > 3$, the $p$-modulus of a Moser family of curves is at least $c_{p} > 0$., 18 pages, 3 figures. v2: Improved main result (replaced 4 by 3)

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

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

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

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 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.

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 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.

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 ∎

We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for every k∈ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then \(|A|\leqq 2^{-ck/{(\log k)}^{2}}N\), for some absolute constant c>0, provided that N is large enough.

Let 1 6 p 1. We write μB,p,n for the probability measure in R with density 1Bn p , where B n p := {x ∈ R : ‖x‖p 6 bp,n} and |B p | = 1. Let μ1, · · · , μk1 be one-dimensional log-concave measures, let pj ∈ [1,∞] and let nj , 1 6 j 6 k2 be positive integers with k1 + ∑k2 j=1 nj = N > n. We show that, for every F ∈ GN,n, L πF (( ⊗1 i=1μi ) ⊗ ( ⊗2 j=1μB,pj,nj )) 6 C, where C > 0 is an absolute constant, Lμ stands for the isotropic constant of μ and πF (μ) denotes the marginal of μ on F .

Let each of [superscript n] particles starting at the origin in Z[superscript 2] perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of [superscript n] occupied sites is (with high probability) close to a disk B [subscript r] of radius r=√n/pi. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant [superscript C] such that with probability [superscript 1], B [subscript r - C log r] C A(pi r[superscript 2]) C B [subscript r+ C log r] for all sufficiently large r., National Science Foundation (U.S.) (grant DMS-1069225), National Science Foundation (U.S.) (grant DMS-0645585), National Science Foundation (U.S.) (Postdoctoral Research Fellowship)

A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov \cite{KC1,KC2} led to an integral operator $K$ -- see (1), (2). They guessed that the equation $E_0(a,T)=1$ where $E_0$ is the largest eigenvalue of the operator $K$ has a solution $T(a)=1-\tau(a)$ with $\tau (a) \sim a^{2/5}$ when $a$ goes to 0. $\tau(a)$ imitates the shift of critical (instability) temperature. We give a rigorous analysis of an anisotropic integral operator $K$ and prove the asymptotic ($*$) -- see Theorem 8 and Proposition 10. Additive Uncertainty Principle (of Landau-Pollack-Slepian [SP], \cite{LP1,LP2}) plays important role in this analysis.