Combinatorial aspects of geometry and number theory  Page description

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Details of project

  
Identifier
43631
Type K
Principal investigator Károlyi, Gyula
Title in Hungarian A geometria és a számelmélet kombinatorikus aspektusai
Title in English Combinatorial aspects of geometry and number theory
Panel Mathematics and Computing Science
Department or equivalent Department of Algebra and Number Theory (Eötvös Loránd University)
Participants Gyarmati, Katalin
Solymosi, Jozsef
Starting date 2003-01-01
Closing date 2007-04-30
Funding (in million HUF) 4.027
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
A kutatás időtartama alatt 55 dolgozatunk született, melyek terjedelemre nézve egyenletesen oszlanak meg a pályazat három résztvevője között. Szinte mindegyik munka igen nívós nemzetközi folyóiratban látott napvilágot, vagy van megjelenés alatt. Nemzetközi viszonylatban is igen jelentősek Gyarmati Katalin pszeudovéletlen sorozatokkal, illetve diofantikus problémákkal foglalkozó munkái, valamint Károlyi Gyula összeghalmazokra vonatkozó struktúratételei, elsősorban az Erdős-Heilbronn problémával kapcsolatban, melyekben a kombinatorikus nullhelytétel és a csoportbővítések elméletének segítségével ér el régóta várt eredményeket az additív kombinatorikában. Legkiemelkedőbbek azonban Solymosi Józsefnek igen eredeti geometriai, számelméleti és kombinatorikus gondolatokat ötvöző dolgozatai, melyekben nehéz Erdős problémákkal valamint Szemerédi tételéhez kapcsolódóan mutat fel olyan eredményeket, amelyekre már a Fields érmes Bourgain, Gowers és Tao is hivatkoznak.
Results in English
In this project we have written 55 papers. The contribution of each of the three participants amounts to roughly 200 pages published mostly in international journals of high standard and reputation. Most significant are the papers of Katalin Gyarmati in which she studies diophantine problems and pseudorandom sequences, the work of Gyula Károlyi in additive combinatorics who with the help of the Combinatorial Nullstellensatz and the theory of group extensions obtained long anticipated structural results in the theory of set addition, in particular in connection with the Erdős-Heilbronn problem, and the results of József Solymosi related to hard Erdős problems resp. Szemerédi's theorem, obtained by very surprising and original combination of combinatorial, geometric and number theoretical ideas, already used and cited by Fields laureates Bourgain, Gowers and Tao.
Full text http://real.mtak.hu/1165/
Decision
Yes





 

List of publications

  
Devillers O; Hurtado F; Károlyi Gy; Seara C: Chromatic variants of the Erdős-Szekeres theorem on points in convex position, Comput Geom Th Appl 26: 193-208, 2003
Solymosi J: Note on integral distances, Discrete Comput Geom 30: 337-342, 2003
Pach J; Solymosi J; Tóth G: Unavoidable configurations in complete topological graphs, Discrete Comput Geom 30: 311-320, 2003
Solymosi J; Vu V: Distinct distances in homogeneous sets, Proc 19th ACM Symp Comp Geom San Diego: 104-105, 2003
Solymosi J: Note on a generalization of Roth’s theorem, In: Aronov B et al. (eds.) Discrete and Computational Geometry, Alg Combin 25, Springer, Berlin, pp. 825-827, 2003
Károlyi Gy: On restricted set addition in Abelian groups, Ann Univ Sci Budapest, Sect Math 46: 47-54, 2003
Károlyi Gy: The Erdős-Heilbronn problem in Abelian groups, Israel J Math 139: 349-359, 2004
Bisztriczky T; Hosono K; Károlyi Gy; Urabe M: Constructions from empty polygons, Period Math Hungar 49: 1-8, 2004
Solymosi J: Note on a question of Erdős and Graham, Combin Prob Comput 13: 263-267, 2004
Gyarmati K: On a family of pseudorandom binary sequences, Period Math Hungar 49: 45-63, 2004
Nesetril J; Solymosi J; Valtr P: A Ramsey property of planar graphs, In: Pach J (ed.) Towards a Theory of Geometric Graphs, Contemp Math 342, AMS, pp. 169-176, 2004
Gyarmati K: An inequality between the measures of pseudorandomness, Ann Univ Sci Budapest, Sect Math 46: 157-166, 2003
Solymosi J; Vu V: Distinct distances in high dimensional homogeneous sets, In: Pach J (ed.) Towards a Theory of Geometric Graphs, Contemp Math 342, AMS, pp. 259-268, 2004
Károlyi Gy; Rosta V: On geometric graph Ramsey numbers, Közlésre benyújtva, Graphs Combin, 2007
Károlyi Gy; Pál A: The cyclomatic number of connected graphs without solvable orbits, Közlésre benyújtva, Advances Math, 2007
Bugeaud Y; Gyarmati K: On generalizations of a problem of Diophantus, Illinois J Math 48: 1105-1115, 2004
Gyarmati K: A polynomial extension of a problem of Diophantus, Publ Math Debrecen 66: 389-405, 2005
Gyarmati K: On a fast version of a pseudorandom generator, In: General Theory of Information Transfer and Combinatorics, Lecture Notes Comp Sci 4123, Springer, Heidelberg, pp. 326-342, 2006
Dietman R; Elsholtz C; Gyarmati K; Simonivits M: Shifted products that are coprime pure powers, J Combin Th A 111: 24-36, 2005
Solymosi J: On sum-sets and product-sets of complex numbers, J Th Nombres Bordeaux 17: 921-924, 2005
Gyarmati K: On the correlation of binary sequences, Studia Sci Math Hungar 42: 79-93, 2005
Károlyi Gy; Lippner G; Valtr P: Empty convex polygons in almost convex sets, Közlésre elfogadva, Period Math Hungar, 2005
Károlyi Gy: A note on the Hopt-Stiefel function, European J Combin 27: 1135-1137, 2006
Károlyi Gy; Solymosi J: Erdős-Szekeres theorem with forbidden order types, J Combin Th A 113: 455-465, 2006
Solymosi J: Arithmetic progressions in sets with small sumsets, Combin Prob Comput 15: 597-603, 2006
Solymosi J: On the number of sums and products, Bull London Math Soc 37: 491-494, 2005
Solymosi J; Vu V: Near optimal bounds for the number of distinct distances in high dimensions, Közlésre elfogadva, Combinatorica, 2005
Gyarmati K; Pethö A; Sárközy A: On linear recursion and pseudorandomness, Acta Arith 118: 359-374, 2005
Gyarmati K: Pseudorandom sequences constructed by the power generator, Period Math Hungar 52: 9-26, 2006
Gyarmati K: On the number of divisors which are values of a polynomial, Közlésre elfogadva, Ramanujan J, 2006
Károlyi Gy: An inverse theorem for the restricted set addition in abelian groups, J Algebra 290: 557-593, 2005
Károlyi Gy: Cauchy-Davenport theorem in group extensions, L'Enseign Math 51: 239-254, 2005
Cooper J N; Solymosi J: Collinear points in permutations, Ann Combin 9: 169-175, 2005
Adamy U; Hoffman M; Solymosi J; Stojakovic M: Coloring octrees, Theoret Comput Sci 363: 11-17, 2006
Solymosi J: Dense arrangements are locally very dense. I, SIAM J Discrete Math 20: 623-627, 2006
Solymosi J; Tóth Cs D: On distinct distances in homogeneous sets in the Euclidean space, Discrete Comput Geom 35: 537-549, 2006
Károlyi Gy; Keleti T; Kós G; Ruzsa I Z: Periodic decomposition of integer valued functions, Közlésre elfogadva, Acta Math Hungar, 2007
Károlyi Gy: Restricted set addition: The exceptional case of the Erdős-Heilbronn conjecture, Közlésre benyújtva, J Combin Th A, 2007
Graham R; Solymosi J: Monochromatic equilateral right triangles on the integer grid, In: Thomas R et al. (eds.) Topics in Discrete Mathematics, Alg Combin 26, Springer, Berlin, pp. 129-132, 2006
Gyarmati K: A note to the paper ''On a fast version of a pseudorandom generator'', Közlésre elfogadva, Ann Univ Sci Budapest, Sect Math, 2006
Gyarmati K; Stewart C L: On powers in shifted products, Közlésre elfogadva, Glasnik Mat, 2006
Gyarmati K; Konyagin S; Ruzsa I Z: Double and triple sums modulo a prime, Közlésre elfogadva, In: Granville A et al. (eds.) Additive Combinatorics, 2006
Gyarmati K; Sárközy A: Equations in finite fields with restricted solution sets. I (Character sums), Közlésre elfogadva, Acta Math Hungar, 2006
Gyarmati K; Sárközy A: Equations in finite fields with restricted solution sets. II (Algebraic equations), Közlésre benyújtva, Acta Math Hungar, 2007
Csikvári P; Gyarmati K; Sárközy A: Density and Ramsey type results on algebraic equations with restricted solution sets, Közlésre benyújtva, J Combin Th A, 2007
Chang M-C; Solymosi J: Sum-product theorems and incidence geometry, J Eur Math Soc 9: 545-560, 2007
Solymosi J: Elementary methods in additive combinatorics, Közlésre elfogadva, In: Granville A et al. (eds.) Additive Combinatorics, 2007
Solymosi J; Tardos G: On the number of k-rich transformations, Proc 23rd ACM Symp Comp Geom Gyeongju: 227-231, 2007
Laba I; Solymosi J: Incidence theorems for pseudoflats, Discrete Comput Geom 37: 163-174, 2007
Solymosi J; Tóth Cs D: Note on a question of Bourgain about geometric incidences, Közlésre benyújtva, Combin Prob Comput, 2007
Solymosi J; Swanepoel K J: Elementary incidence theorems for complex numbers and quaternions, Közlésre benyújtva, SIAM J Discrete Math, 2007
Hart D; Iosevich A; Solymosi J: Sum-product estimates in finite fields via Kloosterman sums, Közlésre elfogadva, Internat Math Res Not, 2006
Károlyi Gy: Balanced subset sums in dense sets of integers, Közlésre benyújtva, Integers. Electr J Combin Number Th, 2007
Károlyi Gy: A compactness argument in the additive theory and the polynomial method, Discrete Math 302: 124-144, 2005
Solymosi J: Regularity, uniformity, and quasirandomness, Proc Nat Acad Sci USA 102: 8075-8076, 2005



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