Corners theorem

Corners theorem In arithmetic combinatorics, the corners theorem states that for every {displaystyle varepsilon >0} , for large enough {stile di visualizzazione N} , any set of at least {displaystyle varepsilon N^{2}} points in the {displaystyle Ntimes N} grid {stile di visualizzazione {1,ldot ,N}^{2}} contains a corner, cioè., a triple of points of the form {stile di visualizzazione {(X,y),(x+h,y),(X,y+h)}} insieme a {displaystyle hneq 0} . It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem.[1] In 2003, József Solymosi gave a short proof using the triangle removal lemma.[2] Contenuti 1 Dichiarazione 2 Proof overview 3 Limiti quantitativi 4 Multidimensional extension 4.1 Multidimensional Szemerédi's Theorem 5 Riferimenti 6 External links Statement Define a corner to be a subet of {displaystyle mathbb {Z} ^{2}} della forma {stile di visualizzazione {(X,y),(x+h,y),(X,y+h)}} , dove {stile di visualizzazione x,y,hin mathbb {Z} } e {displaystyle hneq 0} . Per ogni {displaystyle varepsilon >0} , esiste un numero intero positivo {stile di visualizzazione N(varepsilon )} such that for any {displaystyle Ngeq N(varepsilon )} , any subset {displaystyle Asubseteq {1,ldot ,N}^{2}} with size at least {displaystyle varepsilon N^{2}} contains a corner.

The condition {displaystyle hneq 0} can be relaxed to {displaystyle h>0} by showing that if {stile di visualizzazione A} is dense, then it has some dense subset that is centrally symmetric.

Proof overview What follows is a sketch of Solymosi's argument.

Supponiamo {displaystyle Asubset {1,ldot ,N}^{2}} is corner-free. Construct an auxiliary tripartite graph {stile di visualizzazione G} with parts {stile di visualizzazione X={X_{1},ldot ,X_{N}}} , {displaystyle Y={si_{1},ldot ,si_{N}}} , e {displaystyle Z={z_{1},ldot ,z_{2N}}} , dove {stile di visualizzazione x_{io}} corresponds to the line {displaystyle x=i} , {stile di visualizzazione y_{j}} corresponds to the line {displaystyle y=j} , e {stile di visualizzazione z_{K}} corresponds to the line {displaystyle x+y=k} . Connect two vertices if the intersection of their corresponding lines lies in {stile di visualizzazione A} .

Note that a triangle in {stile di visualizzazione G} corresponds to a corner in {stile di visualizzazione A} , except in the trivial case where the lines corresponding to the vertices of the triangle concur at a point in {stile di visualizzazione A} . It follows that every edge of {stile di visualizzazione G} is in exactly one triangle, so by the triangle removal lemma, {stile di visualizzazione G} ha {stile di visualizzazione o(|V(G)|^{2})} bordi, Così {stile di visualizzazione |UN|=o(N^{2})} , come desiderato.

Quantitative bounds Let {stile di visualizzazione r_{angle }(N)} be the size of the largest subset of {stile di visualizzazione [N]^{2}} which contains no corner. The best known bounds are {stile di visualizzazione {frac {N^{2}}{2^{(c_{1}+o(1)){mq {tronco d'albero _{2}N}}}}}leq r_{angle }(N)leq {frac {N^{2}}{(registro registro N)^{c_{2}}}},} dove {stile di visualizzazione c_{1}ca 1.822} e {stile di visualizzazione c_{2}ca 0.0137} . The lower bound is due to Green,[3] building on the work of Linial and Shraibman.[4] The upper bound is due to Shkredov.[5] Multidimensional extension A corner in {displaystyle mathbb {Z} ^{d}} is a set of points of the form {stile di visualizzazione {un}tazza {a+he_{io}:1leq ileq d}} , dove {stile di visualizzazione e_{1},ldot ,e_{d}} is the standard basis of {displaystyle mathbb {R} ^{d}} , e {displaystyle hneq 0} . The natural extension of the corners theorem to this setting can be shown using the hypergraph removal lemma, in the spirit of Solymosi's proof. The hypergraph removal lemma was shown independently by Gowers[6] and Nagle, Rodl, Schacht and Skokan.[7] Multidimensional Szemerédi's Theorem The multidimensional Szemerédi theorem states that for any fixed finite subset {displaystyle Ssubseteq mathbb {Z} ^{d}} , e per ogni {displaystyle varepsilon >0} , esiste un numero intero positivo {stile di visualizzazione N(S,varepsilon )} such that for any {displaystyle Ngeq N(S,varepsilon )} , any subset {displaystyle Asubseteq {1,ldot ,N}^{d}} with size at least {displaystyle varepsilon N^{d}} contains a subset of the form {displaystyle acdot S+h} . This theorem follows from the multidimensional corners theorem by a simple projection argument.[6] In particolare, Roth's theorem follows directly from the ordinary corners theorem.

References ^ Ajtai, Miklós; Lei appartiene a te, Modificare (1974). "Sets of lattice points that form no squares". Stud. Sci. Matematica. ungherese. 9: 9–11. SIG 0369299.. ^ Solimosi, Joseph (2003). "Note on a generalization of Roth's theorem". In Aronov, Boris; Basu, Saugata; L'odore, John; et al. (eds.). Discrete and computational geometry. Algorithms and Combinatorics. vol. 25. Berlino: Springer-Verlag. pp. 825–827. doi:10.1007/978-3-642-55566-4_39. ISBN 3-540-00371-1. SIG 2038505. ^ Verde, Ben (2021). "Lower Bounds for Corner-Free Sets". arXiv:0710.3032 [math.CO]. ^ Linial, Nati; Shraibman, Adi (2021). "Larger Corner-Free Sets from Better NOF Exactly-N Protocols". Discrete Analysis. 2021. arXiv:2102.00421. doi:10.19086/da.28933. S2CID 231740736. ^ Shkredov, I.D. (2006). "On a Generalization of Szemerédi's Theorem". Atti della London Mathematical Society. 93 (3): 723–760. arXiv:math/0503639. doi:10.1017/S0024611506015991. S2CID 55252774. ^ Salta su: a b Gower, Timoteo (2007). "Regolarità dell'ipergrafo e teorema multidimensionale di Szemerédi". Annali di matematica. 166 (3): 897–946. arXiv:0710.3032. doi:10.4007/annali.2007.166.897. SIG 2373376. S2CID 56118006. ^ Rodl, V.; Nagle, B.; Maglione, J.; Schacht, M.; Kohayakawa, Y. (2005-05-26). "From The Cover: The hypergraph regularity method and its applications". Atti dell'Accademia Nazionale delle Scienze. 102 (23): 8109–8113. Bibcode:2005PNAS..102.8109R. doi:10.1073/pnas.0502771102. ISSN 0027-8424. PMC 1149431. PMID 15919821. External links Proof of the corners theorem on polymath. Categorie: 1974 introductions1974 in mathematicsRamsey theoryAdditive combinatoricsTheorems in combinatorics