Applications of Combinatorics in Multiplicative Number Theory  Page description

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Details of project

 
Identifier
115978
Type PD
Principal investigator Pach, Péter Pál
Title in Hungarian Kombinatorikus eszközök alkalmazása a multiplikatív számelméletben
Title in English Applications of Combinatorics in Multiplicative Number Theory
Keywords in Hungarian sűrűségi tétel, Ramsey-típusú tétel, egyenletek megoldhatósága, multiplikatív Sidon-sorozatok, rainbow-típusú tételek
Keywords in English density theorem, Ramsey-type theorem, equation solvability, multiplicative Sidon-sequences, rainbow-type theorem
Discipline
Mathematics (Council of Physical Sciences)100 %
Ortelius classification: Number theory
Panel Mathematics and Computing Science
Department or equivalent Department of Computer Science and Information Theory (Budapest University of Technology and Economics)
Starting date 2015-09-01
Closing date 2019-08-31
Funding (in million HUF) 20.037
FTE (full time equivalent) 2.33
state running project





 

Final report

 
Results in Hungarian
A legfontosabb elért eredmény a polinom módszer egy új változatának kidolgozása, melynek segítségével megmutattuk, hogy Z_4^n csoport egy 3 hosszú számtani sorozatot nem tartalmazó részhalmazának mérete exponenciálisan kicsi. Ez volt az első ,,exponenciális nyereség'' ilyen típusú problémák esetében. A módszer publikálása óta már számos alkalmazása született, például a cap set probléma és az Erdős-Szemerédi napraforgósejtés megoldása. Egy másik főbb eredmény az x+y=p(z) egyenlet Ramsey-problémájának megoldása az egészek fölött. Továbbá számos eredményt értünk el a multiplikatív Sidon-sorozatok és multiplikatív bázisok területén, mint például a legkisebb multiplikatív bázis méretének meghatározása, Erdős egy kapcsolódó kérdésének megválaszolása, a multiplikatív Sidon-sorozatok leszámlálása, illetve végtelen Sidon-sorozatok lehető legnagyobb aszimptotikus sűrűségének meghatározása.
Results in English
The most important result is developing a new variant of the polynomial method to solve a question in Additive Combinatorics, namely to prove that sets avoiding nontrivial 3-term arithmetic progressions in Z_4^n are exponentially small.  This ``exponential saving'' was the first of a kind for problems of this sort, and till then a large number of applications have been seen including the solutions of the cap set problem and the Erdős-Szemerédi sunflower conjecture. An important achievement is resolving the Ramsey problem of the equation x+y=p(z) over the integers. Finally, we obtained a number of results about multiplicative Sidon sets and multiplicative bases, including determining the smallest possible size of a multiplicative basis, answering a related question of Erdős, enumerating the multiplicative Sidon sets and  studying infinite multiplicative Sidon sets. 
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=115978
Decision
Yes





 

List of publications

 
Croot E, Lev V F, Pach P P: Progression-free sets in Z_4^n are exponentially small, Ann. of Math., 185 (1) 331-337., 2017
Pach P P, Sándor Cs: Multiplicative bases and an Erdős problem, Combinatorica, 38 (5) 1175-1203., 2018
Pach P P: Monochromatic solutions to the equation x+y=z^2 in the interval [N,cN^4], Bulletin of the London Mathematical Society, 50 (6) (2018) 1113-1116., 2018
Liu H, Pach P P: The number of multiplicative Sidon sets of integers, Journal of Combinatorial Theory, Series A, 165 152-175., 2019
Pach P P, Sándor Cs: On infinite multiplicative Sidon sets, European Journal of Combinatorics, 76 (2019) 37-52., 2018
Pach P P: An improved upper bound for the size of the multiplicative 3-Sidon sets, Int. J. Number Theory 15 (8) 1721–1729., 2019
Hegyvári N, Hennecart F, Pach P P: On the density of sumsets and product sets, Australasian Journal of Combinatorics, 74 (1) (2019) 1-16., 2019
Pach P P: Normal forms under Simon’s congruence, Semigroup Forum 97 (2) 251-267., 2018
Liu H, Pach P P, Sándor Cs: Polynomial Schur's theorem, (21 pages), submitted, 2019
Elsholtz C, Pach P P: Caps and progression-free sets in Z_m^n, (43 pages), submitted, 2019
Liu H, Pach P P, Palincza R: The number of maximum primitive sets of integers, (11 pages), submitted, 2019
Caicedo A E, Chartier T A C, Pach P P: Coloring the n-smooth numbers with n colors, (60 pages) submitted, 2019
Pach P P, Sándor Cs: On infinite multiplicative Sidon sets, submitted, 2017
Pach P P: Normal forms under Simon’s congruence, Semigroup Forum 97 (2) 251-267., 2018
Pach P P: Monochromatic solutions to the equation x+y=z^2 in the interval [N,cN^4], Bulletin of the London Mathematical Society, to appear, 2018
Pach P P: Polynomials, Rank and Cap Sets, Joint International Meeting of the Chinese Mathematical Society and the American Mathematical Society, Shanghai, 2018
Pach P P: On multiplicative Sidon sets, Combinatorial and Additive Number Theory, New York, 2018
Pach P P: Polynomial Schur's theorem, Arithmetic Ramsey Theory in Manchester, 2018
Pach P P, Sándor Cs: On infinite multiplicative Sidon sets, European Journal of Combinatorics, to appear, 2018
Pach P P: Polynomials, rank and cap sets, Workshop on Algebraic Methods in Combinatorics, Harvard, 2017
Pach P P: Polynomials, rank and cap sets, 10 Year Anniversary DIMAP Workshop, University of Warwick, 2017
Pach P P: Normal form for the words under Simon’s congruence, submitted, 2016
Pach P P: On some Multiplicative Problems of Erdős, Combinatorial and Additive Number Theory, Graz, 2016
Pach P P: On Multiplicative Bases and some Related Problems, Additive Combinatorcs in Bordeaux, 2016
Croot E, Lev V F, Pach P P: Progression-free sets in Z_4^n are exponentially small, Ann. of Math., 185 (1) 331-337., 2017
Hegyvári N, Hennecart F, Pach P P: On the density of sumsets and product sets, submitted, 2017
Pach P P: Progression-free sets and the polynomial method, 10th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, Budapest, 2017
Pach P P: Polynomials and progression-free sets, CanaDAM, Toronto, 2017
Pach P P: Polynomials and progression-free sets, Journées Arithmétiques, Caen, 2017
Pach P P: On progression-free sets via the polynomial method, Vilnius Conference in Combinatorics and Number Theory, Vilnius, 2017
Pach P P, Sándor Cs: Multiplicative bases and an Erdős problem, Combinatorica, to appear, 2017
Pach P P: Progression-free sets in Z_4^n, Combinatorial and Additive Number Theory, New York, 2016
Pach P P: Számtani sorozatot nem tartalmazó halmazok, Matematikai Lapok 22:(1) 1-7., 2016
Pach P P: Progression-free sets, Strobl, Conference on Elementary and analytic number theory (ELAZ), 2016
Croot E, Lev V F, Pach P P: Progression-free sets in Z_4^n are exponentially small, Ann. of Math., to appear, 2016
Pach P P: Normal form for the words under Simon’s congruence, submitted, 2016
Pach P P: On some Multiplicative Problems of Erdős, Combinatorial and Additive Number Theory, Graz, 2016
Pach P P: On Multiplicative Bases and some Related Problems, Additive Combinatorcs in Bordeaux, 2016
Pach P P, Sándor Cs: Multiplicative bases and an Erdős problem, submitted, 2016
Pach P P: Generalized multiplicative Sidon sets, Journal of Number Theory 157 507-529., 2015
Pach P P: The polynomial method and the cap set problem, One day workshop on Extremal Combinatorics, EWHA University, Seoul, 2019
Pach P P: Polynomial Schur's Theorem, London Colloquia in Combinatorics, 2019
Pach P P: The polynomial method and the cap set problem, FU Berlin, Colloquium, 2019
Pach P P: Polynomials, Rank and Cap Sets, The 11th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications, Tokyo, 2019
Pach P P: On some applications of graph theory to number theoretic problems, IBS One-Day Conference on Extremal Graph Theory, Daejeon, 2019
Pach P P: Polynomials, rank and cap sets, Mathematics Colloquium, TU Graz, 2018
Pach P P: Progression-free sets and rank of matrices, Additive combinatorics, Linz, 2018




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