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2. Beurling integers with lacunarity

Author

Ruzsa, Imre Z.

Subjects
Mathematics - Number Theory, 11N80
Abstract

We present examples of multiplicative semigroups of positive reals (Beurling's generalized integers) with gaps bounded from below., Comment: 10 pages. To be published in Mathematica Pannonica

Published
2023

3. The fractional chromatic number of the plane is at least 4

Author

Matolcsi, Máté, Ruzsa, Imre Z., Varga, Dániel, and Zsámboki, Pál

Subjects
Mathematics - Combinatorics, Mathematics - Metric Geometry
Abstract

We prove that the fractional chromatic number $\chi_f(\mathbb{R}^2)$ of the unit distance graph of the Euclidean plane is greater than or equal to $4$. A fundamental ingredient of the proof is the notion of geometric fractional chromatic number $\chi_{gf}(\mathbb{R}^2)$ introduced recently by Ambrus et al. First, we establish that $\chi_f(\mathbb{R}^2)=\chi_{gf}(\mathbb{R}^2)$ by exploiting the amenability of the group of Euclidean transformations in dimension 2. Second, we provide a specific unit distance graph $G$ on 27 vertices such that $\chi_{gf}(G)=4$. We also provide a natural connection of $\chi_f(\mathbb{R}^2)$ to the maximal size of independent sets in finite unit distance graphs.

Published
2023

4. Additive decomposition of signed prime

Author

Ruzsa, Imre Z.

Subjects
Mathematics - Number Theory, 11P32, 11N99
Abstract

Under the prime-tuple hypothesis, the set of signed primes is a sumset., Comment: 5 pages, submitted to Acta Arithmetica

Published
2022

5. Artificial intelligence for cardiovascular disease risk assessment in personalised framework: a scoping review

Author

Singh, Manasvi, Kumar, Ashish, Khanna, Narendra N., Laird, John R., Nicolaides, Andrew, Faa, Gavino, Johri, Amer M., Mantella, Laura E., Fernandes, Jose Fernandes E., Teji, Jagjit S., Singh, Narpinder, Fouda, Mostafa M., Singh, Rajesh, Sharma, Aditya, Kitas, George, Rathore, Vijay, Singh, Inder M., Tadepalli, Kalyan, Al-Maini, Mustafa, Isenovic, Esma R., Chaturvedi, Seemant, Garg, Deepak, Paraskevas, Kosmas I., Mikhailidis, Dimitri P., Viswanathan, Vijay, Kalra, Manudeep K., Ruzsa, Zoltan, Saba, Luca, Laine, Andrew F., Bhatt, Deepak L., and Suri, Jasjit S.

Published
2024
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6. In-hospital Outcomes of Rotational Atherectomy in ST-Elevation Myocardial Infarction: Results From the Multicentre ROTA-STEMI Network

Author

Hemetsberger, Rayyan, Mankerious, Nader, Muntané-Carol, Guillem, Temporal, Justin, Sulimov, Dmitriy, Gaede, Luise, Woitek, Felix, Grau, Edgar Fadeuilhe, Scalamogna, Maria, Olschewski, Maximilian, Mitsis, Andreas, Ruzsa, Zoltán, Toth, Gabor G., Heyer, Hajo, Toelg, Ralph, Gómez-Hospital, Joan A., Mügge, Andreas, Hengstenberg, Christian, Mangner, Norman, Gori, Tommaso, Cassese, Salvatore, Suárez, Xavier Carrillo, Abdel-Wahab, Mohamed, Johnson, Thomas, Richardt, Gert, and Allali, Abdelhakim

Published
2024
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8. SoutheAsTern eUrope microciRculATION (SATURATION) registry - Design and rationale

Author

Odanovic, Natalija, Tsioufis, Konstantinos, Dimitriadis, Kyriakos, Sakalidis, Athanasios, Papafaklis, Michail I., Davlouros, Periklis, Ivanov, Igor, Cankovic, Milenko, Kalogeropoulos, Andreas S., Hamilos, Michalis, Sideras, Emmanuel, Konigstein, Maayan, Zornitzki, Lior, Kovarnik, Tomas, Ruzsa, Zoltan, Piroth, Zsolt, Zdravkovic, Marija, Mehmedbegovic, Zlatko, Miovski, Zoran, Jurin, Hrvoje, Kanovsky, Jan, Regev, Ehud, Shah, Samit, and Ilic, Ivan

Published
2024
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9. Sumsets of Semiconvex sets

Author

Ruzsa, Imre and Solymosi, Jozsef

Subjects
Mathematics - Combinatorics
Abstract

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset sizes., Comment: To appear in Canad. Math. Bull

Published
2020
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10. On sums and products along the edges, II

Author

Alon, Noga, Ruzsa, Imre, and Solymosi, Jozsef

Subjects
Mathematics - Combinatorics
Abstract

This note is a continuation of an earlier paper by the authors. We describe improved constructions addressing a question of Erd\H{o}s and Szemer\'edi on sums and products of real numbers along the edges of a graph. We also add a few observations about related versions of the problem.

Published
2020

11. A Weighted Pr\'ekopa-Leindler inequality and sumsets with quasicubes

Author

Green, Ben, Matolcsi, Dávid, Ruzsa, Imre, Shakan, George, and Zhelezov, Dmitrii

Subjects
Mathematics - Number Theory, Mathematics - Combinatorics
Abstract

We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Pr\'ekopa-Leindler inequality. This is then applied to show that if $A, B \subseteq \mathbb{Z}^d$ are finite sets and $U$ is a subset of a "quasicube" then $|A + B + U| \geq |A|^{1/2} |B|^{1/2} |U|$. This result is a key ingredient in forthcoming work of the fifth author and P\"alv\"olgyi on the sum-product phenomenon., Comment: 5 pages

Published
2020

12. An analytic approach to cardinalities of sumsets

Author

Matolcsi, Dávid, Ruzsa, Imre, Shakan, George, and Zhelezov, Dmitrii

Subjects
Mathematics - Number Theory, Mathematics - Combinatorics
Abstract

Let $d$ be a positive integer and $U \subset \mathbb{Z}^d$ finite. We study $$\beta(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$\beta(U) = |U|,$$ whenever $U$ is a subset of $\{0,1\}^d$. Our methods parallel those used for the Pr\'ekopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality., Comment: 25 pages

Published
2020

13. On distinct consecutive differences

Author

Ruzsa, Imre, Shakan, George, Solymosi, Jozsef, and Szemerédi, Endre

Subjects
Mathematics - Combinatorics
Abstract

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to the constant., Comment: This paper supersedes arXiv:math/0503069

Published
2019

15. Triangulations and a discrete Brunn-Minkowski inequality in the plane

Author

Böröczky, Károly J., Matolcsi, Máté, Ruzsa, Imre Z., Santos, Francisco, and Serra, Oriol

Subjects
Mathematics - Combinatorics
Abstract

For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjecture the following analogue of the Brunn-Minkowski inequality: for any two point sets $A,B \subset {\mathbb R}^2$ one has \[ {\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}. \] We prove this conjecture in several cases: if $[A]=[B]$, if $B=A\cup\{b\}$, if $|B|=3$, or if none of $A$ or $B$ has interior points., Comment: 30 pages

Published
2018
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16. Nearly subadditive sequences

Author

Furedi, Zoltan and Ruzsa, Imre Z.

Subjects
Mathematics - Combinatorics, 40A05, 11K65, 05A16
Abstract

We show that the de Bruijn-Erd\H{o}s condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\leq f(1)\leq f(2)\leq f(3)\leq \dots $ such that \begin{equation}\sum_{ n=1}^{\infty} f(n)/n^2 = \infty. \end{equation} Then, there exists a sequence $\{b(n)\}_{n=1,2,\dots}$ satisfying \begin{equation}\label{eq1} b(n+m) \leq b(n) + b(m) + f(n+m) \end{equation} such that the sequence of slopes $\{ b(n)/n\}_{n=1,2,\dots}$ takes every rational number. When the series is bounded we improve their result as follows. If there exist $N$ and real $\mu >1$ such that near $f$-subadditivity holds for all pairs $(n,m)$ with $N\leq n\leq m \leq \mu n$, then $\lim_n b(n)/n $ exists., Comment: 11 pages

Published
2018

17. Sums, products and ratios along the edges of a graph

Author

Alon, Noga, Ruzsa, Imre, and Solymosi, Jozsef

Subjects
Mathematics - Combinatorics, Mathematics - Number Theory
Abstract

In their seminal paper Erd\H{o}s and Szemer\'edi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erd\H{o}s-Szemer\'edi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets and ratio sets along the edges of graphs.

Published
2018

18. On the size of the set $AA+A$

Author

Roche-Newton, Oliver, Ruzsa, Imre Z., Shen, Chun-Yen, and Shkredov, Ilya D.

Subjects
Mathematics - Combinatorics
Abstract

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.

Published
2018
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21. Eight pruning deep learning models for low storage and high-speed COVID-19 computed tomography lung segmentation and heatmap-based lesion localization: A multicenter study using COVLIAS 2.0

Author

Agarwal, Mohit, Agarwal, Sushant, Saba, Luca, Chabert, Gian Luca, Gupta, Suneet, Carriero, Alessandro, Pasche, Alessio, Danna, Pietro, Mehmedovic, Armin, Faa, Gavino, Shrivastava, Saurabh, Jain, Kanishka, Jain, Harsh, Jujaray, Tanay, Singh, Inder M., Turk, Monika, Chadha, Paramjit S., Johri, Amer M., Khanna, Narendra N., Mavrogeni, Sophie, Laird, John R., Sobel, David W., Miner, Martin, Balestrieri, Antonella, Sfikakis, Petros P., Tsoulfas, George, Misra, Durga Prasanna, Agarwal, Vikas, Kitas, George D., Teji, Jagjit S., Al-Maini, Mustafa, Dhanjil, Surinder K., Nicolaides, Andrew, Sharma, Aditya, Rathore, Vijay, Fatemi, Mostafa, Alizad, Azra, Krishnan, Pudukode R., Yadav, Rajanikant R., Nagy, Frence, Kincses, Zsigmond Tamás, Ruzsa, Zoltan, Naidu, Subbaram, Viskovic, Klaudija, Kalra, Manudeep K., and Suri, Jasjit S.

Published
2022
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22. On the arithmetic Kakeya conjecture of Katz and Tao

Author

Green, Ben and Ruzsa, Imre

Subjects
Mathematics - Number Theory, Mathematics - Classical Analysis and ODEs, Mathematics - Combinatorics
Abstract

The arithmetic Kakeya conjecture, formulated by Katz and Tao in 2002, is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in $\mathbb{R}^n$ is $n$. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds., Comment: 22 pages

Published
2017

23. Average Goldbach and the Quasi-Riemann Hypothesis

Author

Bhowmik, Gautami and Ruzsa, Imre Z.

Subjects
Mathematics - Number Theory, 11P32, 11M26
Abstract

We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an equivalence between such asymptotics and the Riemann Hypothesis.

Published
2017

24. Convex sequences may have thin additive bases

Author

Ruzsa, Imre Z. and Zhelezov, Dmitrii

Subjects
Mathematics - Combinatorics, 11B13
Abstract

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

Published
2017
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28. Monochromatic paths for the integers

Author

Guerreiro, João, Ruzsa, Imre Z., and Silva, Manuel

Subjects
Mathematics - Combinatorics
Abstract

Recall that van der Waerden's theorem states that any finite coloring of the naturals has arbitrarily long monochromatic arithmetic sequences. We explore questions about the set of differences of those sequences.

Published
2016
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29. More differences than multiple sums

Author

Ruzsa, Imre Z.

Subjects
Mathematics - Number Theory, Mathematics - Combinatorics, 11B13, 11B75, 05B10
Abstract

We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.

Published
2016

30. Pembrolizumab alone or combined with chemotherapy versus chemotherapy as first-line therapy for advanced urothelial carcinoma (KEYNOTE-361): a randomised, open-label, phase 3 trial

Author

Cascallar, Diana Vera, Varela, Mirta, Lazzaro, Mauricio Fernandez, Kaen, Diego Lucas, Gatica, Gabriela, Flores, David Hugo, Falco, Agustin, Molina, Matias, Van Aelst, Filip, Vulsteke, Christof, Sautois, Brieuc, Machiels, Jean-Pascal, Schallier, Denis, Brust, Leandro, Rapatoni, Liane, Azevedo, Sergio J, Marinho, Gisele, Soares, Joao Paulo Holanda, Dzik, Carlos, Almeida Silva, Jamile, Fay, Andre Poisl, Gingerich, Joel, Fradet, Yves, Ferrario, Cristiano, Potvin, Kylea, Vanhuyse, Marie, Abdelsalam, Mahmoud, Cheng, Susanna, Caglevic, Christian, Reyes, Felipe, Leal, Jose Luis, Francisco, Francisco, Ibanez, Carolina, Joly, Florence, Laguerre, Brigitte, Ladoire, Sylvain, Flechon, Aude, Topart, Delphine, Huillard, Olivier, Oudard, Stéphane, Gross-Goupil, Marine, Culine, Stephane, Loriot, Yohann, Gravis, Gwenaelle, Reichardt, Peter, Retz, Margitta, Herden, Jan, Pfister, David, Ohlman, Carsten, Stoeckle, Michael, Wirth, Manfred, Lorch, Anja, Niegisch, Guenter, Goebell, Peter J, Boegemann, Martin, Merseburger, Axel, Gakis, Georgios, Bedke, Jens, Neisius, Andreas, Thomas, Christian, Hoefner, Thomas, Telekes, Andras, Kosa, Judit Erzsebet, Revesz, Janos, Bodoky, Gyorgy, Csoszi, Tibor, Csejtei, Andras, Geczi, Lajos, Ruzsa, Agnes, Kolonics, Zsuzsanna, Erfan, Jozsef, McDermott, Ray, Bambury, Richard, Sella, Avishay, Frank, Stephen Jay, Kejzman, Daniel, Goldman, Olesya, Rosenbaum, Eli, Peer, Avivit, Berger, Raanan, Rouvinov, Keren, Sarid, David, Fukasawa, Satoshi, Arai, Gaku, Yamaguchi, Akito, Yokomizo, Akira, Takayama, Tatsuya, Kinoshita, Hidefumi, Kikuchi, Eiji, Mizuno, Ryuichi, Fujii, Yasuhisa, Sassa, Naoto, Matsukawa, Yoshihisa, Fujimoto, Kiyohide, Matsubara, Nobuaki, Tanikawa, Toshiki, Tomita, Yoshihiko, Nishimura, Kazuo, Tsujihata, Masao, Oyama, Masafumi, Masumori, Naoya, Kanayama, Hiroomi, Takano, Toshimi, Miura, Yuji, Miyazaki, Jun, Joraku, Akira, Kimura, Tomokazu, Yamamoto, Yoshiaki, Kobayashi, Kazuki, De Wit, Ronald, Aarts, Maureen, Gerritsen, Winald, Los, Maartje, Beerepoot, Laurens, Izmailov, Adel, Gorelov, Sergey Igorevich, Alekseev, Boris Yakovlevich, Semenov, Andrey, Kostorov, Vladimir Anatolyevich, Alekseev, Sergey M, Zyryanov, Alexander, Oschepkov, Vasiliy Nikolaevich, Shidin, Vladimir Aleksandrovich, Vladimirov, Vladimir Ivanovich, Gafanov, Rustem Airatovich, Karlov, Petr Alexandrovich, Anderson, David Brian, Shepherd, Lucinda, Cohen, Graham Lawrence, Rapoport, Bernardo Louis, Ruff, Paul, Lee, Nari, Bae, Woo Kyun, Lee, Hyo Jin, Herranz, Urbano Anido, Rodriguez-Vida, Alejo, Morales Barrera, Rafael, Grande, Enrique, Alonso Gordoa, Teresa, Guma Padro, Josep, Gauna, Daniel Castellano, Arranz, Jose Angel, Munoz Langa, Jose, Sarrio, Regina Girones, Montesa Pino, Alvaro, Juan Fita, Maria Jose, Su, Yu-Li, Lin, Yung-Chang, Su, Wen-Pin, Shen, Ying-Chun, Chang, Yen-Hwa, Huang, Yi-Hsiu, Sriuranpong, Virote, Chansriwong, Phichai, Srimuninnimit, Vichien, Danchaivijitr, Pongwut, Abali, Huseyin, Yavuz, Sinan, Ozyilkan, Ozgur, Sendur, Mehmet Ali Nahit, Ekenel, Meltem, Ozguroglu, Mustafa, Arslan, Cagatay, Gunduz, Seyda, Ozdogan, Mustafa, Birtle, Alison, Powles, Thomas, Huddart, Robert, de Santis, Maria, Zarkar, Anjali, Evans, Linda, Hussain, Syed, DiSimone, Christopher, Muina, Antonio F, Schlegel, Peter, Jhangiani, Haresh S, Harrison, Michael, Slater, Dennis E, Wright, David, Percent, Ivor J, Lin, Jianqing, Hwang, Clara, Mamtani, Ronac, Gupta, Sumati, Bajaj, Madhuri, Galamaga, Robert, Eklund, John, Wallace, James, Shtivelband, Mikhail, Suh, Jason Jung-Gon, Burhani, Nafisa, Eadens, Matthew, Gunturu, Krishna, Burgess, Earle, Wong, John, Chaudhry, Arvind, Van Veldhuizen, Peter, Graff, Stephanie, Thomas, Christian A, Schnadig, Ian D, Carneiro, Benedito, Hussain, Maha, Morgans, Alicia, Fitzharris, John T, Oliff, Ira A, Vuky, Jacqueline, Hauke, Ralph, Baron, Ari, Joshi, Monika, Bolemon, Britt H, Jiang, Peter, Mega, Anthony E, Markus, Maurice, Pfanzelter, Nicklas, Lawler, William Eyre, Cobb, Patrick Wayne, Courtright, Jay G, Jain, Sharad, Doshi, Gurjyot, Gunuganti, Vijay K, Sartor, Oliver Alton, Cole, Scott W, Babiker, Hani, Uchio, Edward M, Drakaki, Alexandra, Mannuel, Heather D, Alva, Ajjai, Guancial, Elizabeth, Fung, Chunkit, Charles, Anthony, Amato, Robert J, Arriaga, Yull, Bowman, Isaac, Ades, Steven, Dreicer, Robert, Yu, Evan, Quinn, David I, Fleming, Mark, Csőszi, Tibor, Özgüroğlu, Mustafa, Géczi, Lajos, Cheng, Susanna Y-S, Oudard, Stephane, Fléchon, Aude, Yu, Evan Y, Nam, Kijoeng, Imai, Kentaro, and Homet Moreno, Blanca

Published
2021
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32. The independence number of a subset of an abelian group

Author

Bajnok, Béla and Ruzsa, Imre

Subjects
Mathematics - Number Theory, 11B75
Abstract

We call a subset $A$ of the (additive) abelian group $G$ {\it $t$-independent} if for all non-negative integers $h$ and $k$ with $h+k \leq t$, the sum of $h$ (not necessarily distinct) elements of $A$ does not equal the sum of $k$ (not necessarily distinct) elements of $A$ unless $h=k$ and the two sums contain the same terms in some order. A {\it weakly $t$-independent} set satisfies this property for sums of distinct terms. We give some exact values and asymptotic bounds for the size of a largest $t$-independent set and weakly $t$-independent set in abelian groups, particularly in the cyclic group ${\mathbb Z}_n$.

Published
2015

33. Exact additive complements

Author

Ruzsa, Imre Z.

Subjects
Mathematics - Number Theory, 11B13, 11B34
Abstract

Let $A,B$ be sets of positive integers such that $A+B$ contains all but finitely many positive integers. S\'ark\"ozy and Szemer\'edi proved that if $ A(x)B(x)/x \to 1$, then $A(x)B(x)-x \to \infty $. Chen and Fang considerably improved S\'ark\"ozy and Szemer\'edi's bound. We further improve their estimate and show by an example that our result is nearly best possible.

Published
2015

34. Carries and the arithmetic progression structure of sets

Author

Monopoli, Francesco and Ruzsa, Imre Z.

Subjects
Mathematics - Number Theory
Abstract

If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 + a_2 \equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also when addition is done modulo $m^2$, with $A$ chosen as a set of coset representatives for the cyclic group $\mathbb{Z}/m \mathbb{Z} \subseteq \mathbb{Z}/m^2\mathbb{Z}$. It is a natural to look for sets $A$ which minimize the number of different carries. In a recent paper, Diaconis, Shao and Soundararajan proved that, when $m=p$, $p$ prime, the only set $A$ which induces two distinct carries, i. e. with $A+A \subseteq \{ x, y \}+A$ for some $x, y \in \mathbb{Z}/p^2\mathbb{Z}$, is the arithmetic progression $[0, p-1]$, up to certain linear transformations. We present a generalization of the result above to the case of generic modulus $m^2$, and show how this is connected to the uniqueness of the representation of sets as a minimal number of arithmetic progression of same difference.

Published
2015

40. Better bounds for planar sets avoiding unit distances

Author

Keleti, Tamás, Matolcsi, Máté, Filho, Fernando Mário de Oliveira, and Ruzsa, Imre Z.

Subjects
Mathematics - Metric Geometry, 42B05, 52C10, 52C17, 90C05
Abstract

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erd\H{o}s asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$., Comment: 16 pages, 1 figure. Contains a Sage script called dstverify.sage, to verify the application of Theorem 3.3. Download the article source to get the script

Published
2014
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42. Difference sets and frequently hypercyclic weighted shifts

Author

Bayart, Frédéric and Ruzsa, Imre

Subjects
Mathematics - Functional Analysis
Abstract

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on $\ell^p(\mathbb Z)$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $\mathcal U$-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on $c_0$. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

Published
2013
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43. Squares and difference sets in finite fields

Author

Bachoc, Christine, Ruzsa, Imre Z., and Matolcsi, Mate

Subjects
Mathematics - Combinatorics, Mathematics - Number Theory, 05C69, 11T06
Abstract

For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset \ZZ_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the "trivial" bound $|B|\leq \sqrt{p}$ we prove $|B|\leq \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for approximately three quarters of the primes $p=4k+1$., Comment: 6 pages

Published
2013

44. Pembrolizumab alone or with chemotherapy versus cetuximab with chemotherapy for recurrent or metastatic squamous cell carcinoma of the head and neck (KEYNOTE-048): a randomised, open-label, phase 3 study

Author

Lerzo, Guillermo, Tatangelo, Marcelo, Varela, Mirta, Zarba, Juan Jose, Boyer, Michael, Gan, Hui, Gao, Bo, Hughes, Brett, Mallesara, Girish, Rischin, Danny, Taylor, Anne, Burian, Martin, Fuereder, Thorsten, Greil, Richard, Barrios, Carlos Henrique, de Castro Junior, Dalvaro Oliveira, Castro, Gilberto, Franke, Fabio Andre, Girotto, Gustavo, Lima, Iane Pinto Figueiredo, Nicolau, Ulisses Ribaldo, Pinto, Gustavo Dix Junqueira, Santos, Lucas, Victorino, Ana-Paula, Chua, Neil, Couture, Felix, Gregg, Richard, Hansen, Aaron, Hilton, John, McCarthy, Joy, Soulieres, Denis, Ascui, Rodrigo, Gonzalez, Pablo, Villanueva, Luis, Torregroza, Marco, Zambrano, Angela, Holeckova, Petra, Kral, Zdenek, Melichar, Bohuslav, Prausova, Jana, Vosmik, Milan, Andersen, Maria, Gyldenkerne, Niels, Jurgens, Hannes, Putnik, Kadri, Reinikainen, Petri, Gruenwald, Viktor, Laban, Simon, Aravantinos, Gerasimos, Boukovinas, Ioannis, Georgoulias, Vassilis, Psyrri, Amanda, Kwong, Dora, Al-Farhat, Yousuf, Csoszi, Tibor, Erfan, Jozsef, Horvai, Geza, Landherr, Laszlo, Remenar, Eva, Ruzsa, Agnes, Szota, Judit, Billan, Salem, Gluck, Iris, Gutfeld, Orit, Popovtzer, Aron, Benasso, Marco, Bui, Simona, Ferrari, Vittorio, Licitra, Lisa, Nole, Franco, Fujii, Takashi, Fujimoto, Yasushi, Hanai, Nobuhiro, Hara, Hiroki, Matsumoto, Koji, Mitsugi, Kenji, Monden, Nobuya, Nakayama, Masahiro, Okami, Kenji, Oridate, Nobuhiko, Shiga, Kiyoto, Shimizu, Yasushi, Sugasawa, Masashi, Tahara, Makoto, Takahashi, Masanobu, Takahashi, Shunji, Tanaka, Kaoru, Ueda, Tsutomu, Yamaguchi, Hironori, Yamazaki, Tomoko, Yasumatsu, Ryuji, Yokota, Tomoya, Yoshizaki, Tomokazu, Kudaba, Iveta, Stara, Zinaida, Wan Ishak, Wan Zamaniah, Cheah, Soon Keat, Aguilar Ponce, Jose, Gonzalez Mendoza, Rene, Hernandez Hernandez, Carlos, Medina Soto, Francisco, Buter, Jan, Hoeben, Ann, Oosting, S., Suijkerbuijk, Karijn, Bratland, Aase, Brydoey, Marianne, Alvarez, Renzo, Mas, Luis, Caguioa, Priscilla, Querol, John, Regala, Eugenio Emmanuel, Tamayo, Maria Belen, Villegas, Ellie May, Kawecki, Andrzej, Karpenko, Andrey, Klochikhin, Arkadiy, Smolin, Alexey, Zarubenkov, Oleg, Goh, Boon Cher, Cohen, Graham, du Toit, Johanna, Jordaan, Christa, Landers, Gregory, Ruff, Paul, Szpak, Waldemar, Tabane, Neonyana, Brana, Irene, Iglesias Docampo, Lara, Lavernia, Javier, Mesia, Ricard, Abel, Edvard, Muratidu, Valentina, Nielsen, Niels, Cristina, Valerie, Rordorf, Tamara, Rothschild, Sacha, Hong, Ruey-Long, Wang, Hung-Ming, Yang, Muh-Hwa, Yeh, Su-Peng, Yen, Chia-Jui, Ngamphaiboon, Nuttapong, Soparattanapaisarn, Nopadol, Sriuranpong, Virote, Aksoy, Sercan, Cicin, Irfan, Ekenel, Meltem, Harputluoglu, Hakan, Ozyilkan, Ozgur, Harrington, Kevin, Agarwala, Sanjiv, Ali, Haythem, Alter, Robert, Anderson, Daniel, Bruce, Justine, Burtness, Barbara, Campbell, Nicholas, Conde, Miguel, Deeken, John, Edenfield, William, Feldman, Lawrence, Gaughan, Elizabeth, Goueli, Basem, Halmos, Balazs, Hegde, Upendra, Hunis, Brian, Jotte, Robert, Karnad, Anand, Khan, Saad, Laudi, Noel, Laux, Douglas, Martincic, Danko, McCune, Steven, McGaughey, Dean, Misiukiewicz, Krzysztof, Mulford, Deborah, Nadler, Eric, Neupane, Prakash, Nunnink, Johannes, Ohr, James, O'Malley, Meaghan, Patson, Brian, Paul, Doru, Popa, Elizabeta, Powell, Steven, Redman, Rebecca, Rella, Vincent, Rocha Lima, Chaio, Sivapiragasam, Abirami, Su, Yungpo, Sukari, Ammar, Wong, Stuart, Yilmaz, Emrullah, Yorio, Jeffrey, Harrington, Kevin J, Soulières, Denis, de Castro, Gilberto, Jr, Basté, Neus, Bratland, Åse, Hughes, Brett G M, Mesía, Ricard, González Mendoza, René, Roy, Ananya, Zhang, Yayan, Gumuscu, Burak, Cheng, Jonathan D, and Jin, Fan

Published
2019
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46. Ramucirumab with cisplatin and fluoropyrimidine as first-line therapy in patients with metastatic gastric or junctional adenocarcinoma (RAINFALL): a double-blind, randomised, placebo-controlled, phase 3 trial

Author

Luft, Alexander V., Karaseva, Nina A., Kowalyszyn, Rubén Dario, Hernandez, Carlos Alberto, Csoszi, Tibor, De Vita, Ferdinando, Pfeiffer, Per, Sugimoto, Naotoshi, Kocsis, Judit, Csilla, Andràs, Bodoky, Gyorgy, Garnica Jaliffe, Georgina, Protsenko, Svetlana, Madi, Ayman, Wojcik, Elzbieta, Brenner, Baruch, Folprecht, Gunnar, Sarosiek, Tomasz, Peltola, Katriina Johanna, Bono, Peter, Ayala, Hubert, Aprile, Giuseppe, Gerardo, Cardellino Giovanni, Huitzil Melendez, Fidel David, Falcone, Alfredo, Di Costanzo, Francesco, Tehfe, Moustapha, Mineur, Laurent, García Alfonso, Pilar, Obermannova, Radka, Senellart, Hélène, Petty, Russell, Samuel, Leslie, Acs, Peter Istvan, Hussein, Maen Abdelkarim, Nechaeva, Marina N., Erdkamp, F.L.G., Won, Elizabeth, Bendell, Johanna Chock, Gallego Plazas, Javier, Lorenzen, Sylvie, Melichar, Bohuslav, Escudero, Miguel Angel, Pezet, Denis, Phelip, Jean-Marc, Kaen, Diego Lucas, Reeves, James A. Jr, Longo Muñoz, Federico, Madhusudan, Srinivasan, Barone, Carlo, Fein, Luis Enrique, Gomez Villanueva, Angel, Hebbar, Mohamed, Prausova, Jana, Visa Turmo, Laura, Vidal Barrull, Joana, Yilmaz, Mette Karen Nytoft, Beny, Alex, Van Laarhoven, H.M.W., DiCarlo, Brian Anthony, Esaki, Taito, Fujitani, Kazumasa, Geboes, Karen, Geva, Ravit, Kadowaki, Shigenori, Leong, Stephen, Machida, Nozomu, Raj, Moses Sundar, Ramirez Godinez, Francisco Javier, Ruzsa, Agnes, Ford, Hugo, Lawler, William E., Maisey, Nicolas Robert, Petera, Jiri, Shacham-Shmueli, Einat, Sinapi, Isabelle, Yamaguchi, Kensei, Hara, Hiroki, Beck, Joseph Thaddeus, Błasińska-Morawiec, Maria, Villalobos Valencia, Ricardo, Alcindor, Thierry, Bajaj, Madhuri, Berry, Scott, Gomez, Christina Maria, Dammrich, Daniel, Patel, Ravindranath, Taieb, Julien, Ten Tije, A.J., Burkes, Ronald L., Cabanillas, Fernando, Firdaus, Irfan, Chua, Cynthia Coo, Hironaka, Shuichi, Hofheinz, Ralf-Dieter, Lim, Howard J., Nordsmark, Marianne, Piko, Bela, Verma, Udit, Wadsley, Jonathan, Yukisawa, Seigo, Gutiérrez Delgado, Francisco, Denlinger, Crystal S., Kallio, Raija, Pikiel, Joanna, Wojcik-Tomaszewska, Joanna, Brezden-Masley, Christine, Jang, Raymond Woo-Jun, Pribylova, Jana, Sakai, Daisuke, Bartoli, Maria Alejandra, Cats, A., Grootscholten, M.I., Dichmann, Robert Andrew, Hool, Hugo, Shaib, Walid, Tsuji, Akihito, Van den Eynde, Marc, Velez-Cortez, Hector, Asmis, Timothy R., Fuchs, Charles S, Shitara, Kohei, Di Bartolomeo, Maria, Lonardi, Sara, Al-Batran, Salah-Eddin, Van Cutsem, Eric, Ilson, David H, Alsina, Maria, Chau, Ian, Lacy, Jill, Ducreux, Michel, Mendez, Guillermo Ariel, Alavez, Alejandro Molina, Takahari, Daisuke, Mansoor, Wasat, Enzinger, Peter C, Gorbounova, Vera, Wainberg, Zev A, Hegewisch-Becker, Susanna, Ferry, David, Lin, Ji, Carlesi, Roberto, Das, Mayukh, and Shah, Manish A

Published
2019
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47. Pembrolizumab versus methotrexate, docetaxel, or cetuximab for recurrent or metastatic head-and-neck squamous cell carcinoma (KEYNOTE-040): a randomised, open-label, phase 3 study

Author

Acosta-Rivera, Mirelis, Adkins, Douglas R., Aghmesheh, Morteza, Ahn, Myung-Ju, Airoldi, Mario, Aleknavicius, Eduardas, Al-Farhat, Yousuf, Algazi, Alain P., Almokadem, Salah, Alyasova, Anna, Bauman, Jessica R., Benasso, Marco, Berrocal, Alfonso, Bray, Victoria, Burtness, Barbara Ann, Caponigro, Francesco, Castro, Ana, Cescon, Terrence P., Chan, Kelvin, Chaudhry, Arvind, Chauffert, Bruno, Cohen, Ezra, Csoszi, Tibor, De Boer, J.P., Delord, Jean-Pierre, Dietz, Andreas, Dinis, Jose, Dupuis, Charlotte, Digue, Laurence, Erfan, Jozsef, Escobar Alvarez, Yolanda, Evans, Mererid, Fidler, Mary Jo, Forster, Martin David, Friesland, Signe, Ganti, Apar K., Geoffrois, Lionnel, Grant, Cliona, Gruenwald, Viktor, Harrington, Kevin, Hoffmann, Thomas, Horvai, Geza, Inciura, Arturas, Jang, Raymond, Jankowska, Petra, Jimeno, Antonio, Joseph, Mano, Juarez Ramiro, Alejandro, Karaszewska, Boguslawa, Kawecki, Andrzej, Keilholz, Ulrich, Keller, Ulrich, Kim, Sung-Bae, Kocsis, Judit, Kotecki, Nuria, Kozloff, Mark F., Lambea, Julio, Landherr, Laszlo, Lantsukhay, Yuri, Lazarev, Sergey Alexandrovich, Lee, Lip Way, Le Tourneau, Christophe, Licitra, Lisa, Lifirenko, Igor Dmitrievich, Mach, Nicolas, Martincic, Danko, Matorin, Oleg Vladmirovhich, McGrath, Margaret, Machiels, Jean-Pascal, Mehra, Ranee, Misiukiewicz, Krzysztof, Morris, John C., Mufazalov, Fagim Fanisovich, Niu, Jiaxin, Pamoorthy Srinivasan, Devraj, Perez Segura, Pedro, Rauch, Daniel, Ribeiro, Maria Leonor, Rodriguez, Cristina, Rolland, Frederic, Russo, Antonio, Ruzsa, Agnes, Sanches, Frederico, Shin, Sang-Won, Shtiveland, Mikhail, Soulieres, Denis, Soria, Ainara, Specenier, Pol, Szekanecz, Eva, Szota, Judit, van Herpen, Carla M.L., Velez-Cortes, Hector A., Walsh, William V., Wilop, Stefan, Winterhalder, Ralph, Wojtukiewicz, Marek, Wong, Deborah, Zandberg, Dan, Cohen, Ezra E W, Soulières, Denis, Dinis, José, Burtness, Barbara, Zhang, Pingye, Cheng, Jonathan, Swaby, Ramona F, and Harrington, Kevin J

Published
2019
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48. A note on the pyjama problem

Author

Malikiosis, R. D., Matolcsi, M., and Ruzsa, I. Z.

Subjects
Mathematics - Combinatorics, Mathematics - Metric Geometry, 52C15
Abstract

This note concerns the so-called pyjama problem, whether it is possible to cover the plane by finitely many rotations of vertical strips of half-width $\varepsilon$. We first prove that there exist no periodic coverings for $\varepsilon<1/3$. Then we describe an explicit (non-periodic) construction for $\varepsilon =1/3-1/48$. Finally, we use a compactness argument combined with some ideas from additive combinatorics to show that a finite covering exists for $\varepsilon =1/5$. The question whether $\varepsilon$ can be arbitrarily small remains open., Comment: 9 pages

Published
2012
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49. Difference sets and positive exponential sums I. General properties

Author

Matolcsi, Mate and Ruzsa, Imre Z.

Subjects
Mathematics - Combinatorics, Mathematics - Number Theory, 11B75, 11P70
Abstract

We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set $A$ the maximal size of a set whose difference set avoids $A$ will be related to positive exponential sums using frequencies from $A$., Comment: 21 pages, a few remarks added, and one theorem slightly extended

Published
2012

50. Real and complex unbiased Hadamard matrices

Author

Matolcsi, Mate, Ruzsa, Imre Z., and Weiner, Mihaly

Subjects
Mathematics - Combinatorics
Abstract

We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard matrices (MUHs) in any dimension $d$ cannot contain more than one real Hadamard matrix. We also give new proofs of several known structural results in low dimensions, for $d\le 6$., Comment: 13 pages

Published
2012

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