A proof of a sumset conjecture of Erdős

Joel Moreira; Florian K. Richter; Donald Robertson

doi:https://doi.org/10.4007/annals.2019.189.2.4

Abstract

In this paper we show that every set $A \subset \mathbb {N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb {N}$, settling a conjecture of Erd\H os. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.

Show/hide bibliography for this article

[MR0360999] $Go to document$ M. A. Akcoglu and A. del Junco, "Convergence of averages of point transformations," Proc. Amer. Math. Soc., vol. 49, pp. 265-266, 1975.

Show bibtex

@ARTICLE{MR0360999, author = {Akcoglu, M. A. and del Junco, A.}, title = {Convergence of averages of point transformations}, journal = {Proc. Amer. Math. Soc.}, fjournal = {Proceedings of the Amer. Math. Soc.}, volume = {49}, year = {1975}, pages = {265--266}, issn = {0002-9939}, mrclass = {28A65}, mrnumber = {0360999}, mrreviewer = {Michael Lin}, doi = {10.2307/2039829}, url = {https://doi.org/10.2307/2039829}, zblnumber = {0278.28011}, }
[arXiv:1701.07791] $Go to document$ U. Andrews, G. Conant, and I. Goldbring, "Definable sets containing productsets in expansions of groups," J. Group Theory, vol. 22, iss. 1, pp. 63-82, 2019.

Show bibtex

@ARTICLE{arXiv:1701.07791, author = {Andrews, Uri and Conant, Gabriel and Goldbring, Isaac}, title = {Definable sets containing productsets in expansions of groups}, journal = {J. Group Theory}, fjournal = {Journal of Group Theory}, volume = {22}, year = {2019}, number = {1}, pages = {63--82}, issn = {1433-5883}, mrclass = {03 (05 11)}, mrnumber = {3895638}, doi = {10.1515/jgth-2018-0038}, url = {https://doi.org/10.1515/jgth-2018-0038}, zblnumber = {06998938}, }
[Beiglbock11] $Go to document$ M. Beiglböck, "An ultrafilter approach to Jin’s theorem," Israel J. Math., vol. 185, pp. 369-374, 2011.

Show bibtex

@ARTICLE{Beiglbock11, author = {Beiglböck, Mathias}, title = {An ultrafilter approach to {J}in's theorem}, journal = {Israel J. Math.}, fjournal = {Israel Journal of Mathematics}, volume = {185}, year = {2011}, pages = {369--374}, issn = {0021-2172}, mrclass = {11B05 (03E05 03H05 28E05 43A07)}, mrnumber = {2837141}, mrreviewer = {\c{S}. A. Basarab}, doi = {10.1007/s11856-011-0114-5}, url = {https://doi.org/10.1007/s11856-011-0114-5}, zblnumber = {1300.11015}, }
[MR2565535] $Go to document$ M. Beiglböck, V. Bergelson, and A. Fish, "Sumset phenomenon in countable amenable groups," Adv. Math., vol. 223, iss. 2, pp. 416-432, 2010.

Show bibtex

@ARTICLE{MR2565535, author = {Beiglböck, Mathias and Bergelson, Vitaly and Fish, Alexander}, title = {Sumset phenomenon in countable amenable groups}, journal = {Adv. Math.}, fjournal = {Advances in Mathematics}, volume = {223}, year = {2010}, number = {2}, pages = {416--432}, issn = {0001-8708}, mrclass = {11B05 (11B13 11B30 37A45 43A07)}, mrnumber = {2565535}, mrreviewer = {Randall McCutcheon}, doi = {10.1016/j.aim.2009.08.009}, url = {https://doi.org/10.1016/j.aim.2009.08.009}, zblnumber = {1187.43002}, }
[BL85] $Go to document$ A. Bellow and V. Losert, "The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences," Trans. Amer. Math. Soc., vol. 288, iss. 1, pp. 307-345, 1985.

Show bibtex

@ARTICLE{BL85, author = {Bellow, A. and Losert, V.}, title = {The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the Amer. Math. Soc.}, volume = {288}, year = {1985}, number = {1}, pages = {307--345}, issn = {0002-9947}, mrclass = {28D05}, mrnumber = {0773063}, mrreviewer = {U. Krengel}, doi = {10.2307/2000442}, url = {https://doi.org/10.2307/2000442}, zblnumber = {0619.47004}, }
[Bergelson85] $Go to document$ V. Bergelson, "Sets of recurrence of ${\bf Z}^m$-actions and properties of sets of differences in ${\bf Z}^m$," J. London Math. Soc. (2), vol. 31, iss. 2, pp. 295-304, 1985.

Show bibtex

@ARTICLE{Bergelson85, author = {Bergelson, Vitaly}, title = {Sets of recurrence of {${\bf Z}^m$}-actions and properties of sets of differences in {${\bf Z}^m$}}, journal = {J. London Math. Soc. (2)}, fjournal = {Journal of the London Mathematical Society. Second Series}, volume = {31}, year = {1985}, number = {2}, pages = {295--304}, issn = {0024-6107}, mrclass = {28D15 (11B05)}, mrnumber = {0809951}, mrreviewer = {J. B. Robertson}, doi = {10.1112/jlms/s2-31.2.295}, url = {https://doi.org/10.1112/jlms/s2-31.2.295}, zblnumber = {0579.10029}, }
[Bergelson96] $Go to document$ V. Bergelson, "Ergodic Ramsey theory—an update," in Ergodic Theory of ${\bf Z}^d$ Actions (Warwick, 1993–19994), Cambridge Univ. Press, Cambridge, 1996, vol. 228, pp. 1-61.

Show bibtex

@INCOLLECTION{Bergelson96, author = {Bergelson, Vitaly}, title = {Ergodic {R}amsey theory---an update}, booktitle = {Ergodic Theory of {${\bf Z}^d$} Actions (Warwick, 1993--19994)}, series = {London Math. Soc. Lecture Note Ser.}, note = {(M. Pollicott and K. Schmidt, eds.)}, volume = {228}, pages = {1--61}, publisher = {Cambridge Univ. Press, Cambridge}, year = {1996}, mrclass = {28D05 (05A18 05D10 11B25)}, mrnumber = {1411215}, mrreviewer = {Karl Petersen}, zblnumber = {0846.05095}, doi = {10.1017/CBO9780511662812.002}, url = {https://doi.org/10.1017/CBO9780511662812.002}, }
[MR2249261] $Go to document$ V. Bergelson, H. Furstenberg, and B. Weiss, "Piecewise-Bohr sets of integers and combinatorial number theory," in Topics in Discrete Mathematics, Springer, Berlin, 2006, vol. 26, pp. 13-37.

Show bibtex

@INCOLLECTION{MR2249261, author = {Bergelson, Vitaly and Furstenberg, Hillel and Weiss, Benjamin}, title = {Piecewise-{B}ohr sets of integers and combinatorial number theory}, booktitle = {Topics in Discrete Mathematics}, series = {Algorithms Combin.}, volume = {26}, pages = {13--37}, publisher = {Springer, Berlin}, year = {2006}, mrclass = {37A45 (11B05 11B75 28D05 62M10)}, mrnumber = {2249261}, mrreviewer = {Bryna Kra}, doi = {10.1007/3-540-33700-8_2}, url = {https://doi.org/10.1007/3-540-33700-8_2}, zblnumber = {1114.37008}, }
[MR513591] $Go to document$ J. F. Berglund, H. D. Junghenn, and P. Milnes, Compact Right Topological Semigroups and Generalizations of Almost Periodicity, Springer, Berlin, 1978, vol. 663.

Show bibtex

@BOOK{MR513591, author = {Berglund, J. F. and Junghenn, H. D. and Milnes, P.}, title = {Compact Right Topological Semigroups and Generalizations of Almost Periodicity}, series = {Lecture Notes in Math.}, volume = {663}, publisher = {Springer, Berlin}, year = {1978}, pages = {x+243}, isbn = {3-540-08919-5}, mrclass = {22A20 (43A60)}, mrnumber = {0513591}, mrreviewer = {A. T.-M. Lau}, zblnumber = {0406.22005}, doi = {10.1007/BFb0061381}, url = {https://doi.org/10.1007/BFb0061381}, }
[Besicovitch26] $Go to document$ A. S. Besicovitch, "On generalized almost periodic functions," Proc. London Math. Soc. (2), vol. 25, pp. 495-512, 1926.

Show bibtex

@ARTICLE{Besicovitch26, author = {Besicovitch, A. S.}, title = {On generalized almost periodic functions}, journal = {Proc. London Math. Soc. (2)}, fjournal = {Proceedings of the London Mathematical Society. Second Series}, volume = {25}, year = {1926}, pages = {495--512}, issn = {0024-6115}, mrclass = {DML}, mrnumber = {1575297}, doi = {10.1112/plms/s2-25.1.495}, url = {https://doi.org/10.1112/plms/s2-25.1.495}, jfmnumber = {52.0263.01}, }
[Besicovitch55] A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.

Show bibtex

@BOOK{Besicovitch55, author = {Besicovitch, A. S.}, title = {Almost Periodic Functions}, publisher = {Dover Publications, Inc., New York}, year = {1955}, pages = {xiii+180}, mrclass = {42.4X}, mrnumber = {0068029}, zblnumber = {0065.07102}, }
[MR0013443] $Go to document$ H. Bohr and E. Følner, "On some types of functional spaces. A contribution to the theory of almost periodic functions," Acta Math., vol. 76, pp. 31-155, 1945.

Show bibtex

@ARTICLE{MR0013443, author = {Bohr, Harald and Følner, Erling}, title = {On some types of functional spaces. {A} contribution to the theory of almost periodic functions}, journal = {Acta Math.}, fjournal = {Acta Mathematica}, volume = {76}, year = {1945}, pages = {31--155}, issn = {0001-5962}, mrclass = {42.4X}, mrnumber = {0013443}, mrreviewer = {R. H. Cameron}, doi = {10.1007/BF02547156}, url = {https://doi.org/10.1007/BF02547156}, zblnumber = {0061.16201}, }
[Bohr25-1] $Go to document$ H. Bohr, "Zur theorie der fast periodischen funktionen," Acta Math., vol. 45, iss. 1, pp. 29-127, 1925.

Show bibtex

@ARTICLE{Bohr25-1, author = {Bohr, Harald}, title = {Zur theorie der fast periodischen funktionen}, note = {I. Eine verallgemeinerung der theorie der fourierreihen}, journal = {Acta Math.}, fjournal = {Acta Mathematica}, volume = {45}, year = {1925}, number = {1}, pages = {29--127}, issn = {0001-5962}, mrclass = {DML}, mrnumber = {1555192}, doi = {10.1007/BF02395468}, url = {https://doi.org/10.1007/BF02395468}, zblnumber = {50.0196.01}, }
[Bohr25-2] $Go to document$ H. Bohr, "Zur Theorie der Fastperiodischen Funktionen," Acta Math., vol. 46, iss. 1-2, pp. 101-214, 1925.

Show bibtex

@ARTICLE{Bohr25-2, author = {Bohr, Harald}, title = {Zur {T}heorie der {F}astperiodischen {F}unktionen}, note = {II. Zusammenhang der fastperiodischen Funktionen mit Funktionen von unendlich vielen Variabeln; gleichmässige Approximation durch trigonometrische Summen}, journal = {Acta Math.}, fjournal = {Acta Mathematica}, volume = {46}, year = {1925}, number = {1-2}, pages = {101--214}, issn = {0001-5962}, mrclass = {DML}, mrnumber = {1555201}, doi = {10.1007/BF02543859}, url = {https://doi.org/10.1007/BF02543859}, jfmnumber = {51.0212.02}, }
[DGJLLM15] $Go to document$ M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, and K. Mahlburg, "On a sumset conjecture of Erdős," Canad. J. Math., vol. 67, iss. 4, pp. 795-809, 2015.

Show bibtex

@ARTICLE{DGJLLM15, author = {Di Nasso, Mauro and Goldbring, Isaac and Jin, Renling and Leth, Steven and Lupini, Martino and Mahlburg, Karl}, title = {On a sumset conjecture of {E}rd{ő}s}, journal = {Canad. J. Math.}, fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques}, volume = {67}, year = {2015}, number = {4}, pages = {795--809}, issn = {0008-414X}, mrclass = {11B05 (11B13 11P70 28D15 37A45)}, mrnumber = {3361013}, mrreviewer = {Sean Eberhard}, doi = {10.4153/CJM-2014-016-0}, url = {https://doi.org/10.4153/CJM-2014-016-0}, zblnumber = {1365.11008}, }
[MR3410920] $Go to document$ T. Eisner, B. Farkas, M. Haase, and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Springer, Cham, 2015, vol. 272.

Show bibtex

@BOOK{MR3410920, author = {Eisner, Tanja and Farkas, B\'{a}lint and Haase, Markus and Nagel, Rainer}, title = {Operator Theoretic Aspects of Ergodic Theory}, series = {Grad. Texts in Math.}, volume = {272}, publisher = {Springer, Cham}, year = {2015}, pages = {xviii+628}, isbn = {978-3-319-16897-5; 978-3-319-16898-2}, mrclass = {47-02 (37-02 37A30 37A55 47A35)}, mrnumber = {3410920}, mrreviewer = {Idris Assani}, doi = {10.1007/978-3-319-16898-2}, url = {https://doi.org/10.1007/978-3-319-16898-2}, zblnumber = {1353.37002}, }
[Erdos77] P. ErdHos, "Problems and results on combinatorial number theory. III," in Number Theory Day (Proc. Conf., Rockefeller Univ., New York, 1976), Springer, Berlin, 1977, vol. 626, pp. 43-72.

Show bibtex

@incollection{Erdos77, author = {Erd{ő}s, Paul}, title = {Problems and results on combinatorial number theory. {III}}, booktitle = {Number Theory Day ({P}roc. {C}onf., {R}ockefeller {U}niv., {N}ew {Y}ork, 1976)}, pages = {43--72}, series={Lecture Notes in Math.}, volume={626}, publisher = {Springer, Berlin}, year = {1977}, mrclass = {10L10 (10A99)}, mrnumber = {0472752}, mrreviewer = {S. L. G. Choi}, zblnumber = {0368.10002}, }
[Erdos80] P. ErdHos, "A survey of problems in combinatorial number theory," Ann. Discrete Math., vol. 6, pp. 89-115, 1980.

Show bibtex

@ARTICLE{Erdos80, author = {Erd{ő}s, Paul}, title = {A survey of problems in combinatorial number theory}, note = {Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978)}, journal = {Ann. Discrete Math.}, fjournal = {Annals of Discrete Mathematics}, volume = {6}, year = {1980}, pages = {89--115}, mrclass = {10-XX (05Axx)}, mrnumber = {0593525}, mrreviewer = {A. L. Whiteman}, zblnumber = {0448.10002}, }
[Erdos_Graham80] P. ErdHos and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Université de Genève, L’Enseignement Mathématique, Geneva, 1980, vol. 28.

Show bibtex

@BOOK{Erdos_Graham80, author = {Erd{ő}s, Paul and Graham, R. L.}, title = {Old and New Problems and Results in Combinatorial Number Theory}, series = {Monogr. Enseign. Math.}, volume = {28}, publisher = {Université de Genève, L'Enseignement Mathématique, Geneva}, year = {1980}, pages = {128}, mrclass = {10-02 (05-02)}, mrnumber = {0592420}, mrreviewer = {L. C. Eggan}, zblnumber = {0434.10001}, }
[Frantzikinakis15] $Go to document$ N. Frantzikinakis, "Multiple correlation sequences and nilsequences," Invent. Math., vol. 202, iss. 2, pp. 875-892, 2015.

Show bibtex

@ARTICLE{Frantzikinakis15, author = {Frantzikinakis, Nikos}, title = {Multiple correlation sequences and nilsequences}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {202}, year = {2015}, number = {2}, pages = {875--892}, issn = {0020-9910}, mrclass = {37A30 (05D10 11B30 37A05)}, mrnumber = {3418246}, mrreviewer = {Vladimir S. Anashin}, doi = {10.1007/s00222-015-0579-7}, url = {https://doi.org/10.1007/s00222-015-0579-7}, zblnumber = {1358.37019}, }
[MR603625] $Go to document$ H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.

Show bibtex

@BOOK{MR603625, author = {Furstenberg, H.}, title = {Recurrence in Ergodic Theory and Combinatorial Number Theory}, note = {M.~B.~Porter Lectures}, publisher = {Princeton University Press, Princeton, N.J.}, year = {1981}, pages = {xi+203}, isbn = {0-691-08269-3}, mrclass = {28D05 (10K10 10L10 54H20)}, mrnumber = {0603625}, mrreviewer = {Michael Keane}, zblnumber = {0459.28023}, doi = {10.1515/9781400855162}, url = {https://doi.org/10.1515/9781400855162}, }
[MR1958753] $Go to document$ E. Glasner, Ergodic Theory via Joinings, Amer. Math. Soc., Providence, RI, 2003, vol. 101.

Show bibtex

@BOOK{MR1958753, author = {Glasner, Eli}, title = {Ergodic Theory via Joinings}, series = {Math. Surveys Monogr.}, volume = {101}, publisher = {Amer. Math. Soc., Providence, RI}, year = {2003}, pages = {xii+384}, isbn = {0-8218-3372-3}, mrclass = {37A15 (28Dxx 37A25 37A35 37A45 37B99 54H20)}, mrnumber = {1958753}, mrreviewer = {Andrés del Junco}, doi = {10.1090/surv/101}, url = {https://doi.org/10.1090/surv/101}, zblnumber = {1038.37002}, }
[GKR] D. Glasscock, A. Koutsogiannis, and F. K. Richter, Multiplicative combinatorial properties of return time sets in minimal dynamical systems, 2018.

Show bibtex

@MISC{GKR, author = {Glasscock, D. and Koutsogiannis, A. and Richter, F. K.}, title = {Multiplicative combinatorial properties of return time sets in minimal dynamical systems}, year = {2018}, zblnumber = {}, }
[MR1091350] $Go to document$ A. Granville, "A note on sums of primes," Canad. Math. Bull., vol. 33, iss. 4, pp. 452-454, 1990.

Show bibtex

@ARTICLE{MR1091350, author = {Granville, Andrew}, title = {A note on sums of primes}, journal = {Canad. Math. Bull.}, fjournal = {Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques}, volume = {33}, year = {1990}, number = {4}, pages = {452--454}, issn = {0008-4395}, mrclass = {11P32 (11N13)}, mrnumber = {1091350}, mrreviewer = {Kee Wai Lau}, doi = {10.4153/CMB-1990-073-7}, url = {https://doi.org/10.4153/CMB-1990-073-7}, zblnumber = {0691.10039}, }
[GT10-2] $Go to document$ B. Green and T. Tao, "An arithmetic regularity lemma, an associated counting lemma, and applications," in An Irregular Mind, János Bolyai Math. Soc., Budapest, 2010, vol. 21, pp. 261-334.

Show bibtex

@INCOLLECTION{GT10-2, author = {Green, Ben and Tao, Terence}, title = {An arithmetic regularity lemma, an associated counting lemma, and applications}, booktitle = {An Irregular Mind}, series = {Bolyai Soc. Math. Stud.}, volume = {21}, pages = {261--334}, publisher = {J\'{a}nos Bolyai Math. Soc., Budapest}, year = {2010}, mrclass = {11B30 (05D05)}, mrnumber = {2815606}, mrreviewer = {David Conlon}, doi = {10.1007/978-3-642-14444-8_7}, url = {https://doi.org/10.1007/978-3-642-14444-8_7}, zblnumber = {1222.11015}, }
[MR2415379] $Go to document$ B. Green and T. Tao, "The primes contain arbitrarily long arithmetic progressions," Ann. of Math. (2), vol. 167, iss. 2, pp. 481-547, 2008.

Show bibtex

@ARTICLE{MR2415379, author = {Green, Ben and Tao, Terence}, title = {The primes contain arbitrarily long arithmetic progressions}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {167}, year = {2008}, number = {2}, pages = {481--547}, issn = {0003-486X}, mrclass = {11N13 (11A41 11B25 37A45)}, mrnumber = {2415379}, mrreviewer = {Tamar Ziegler}, doi = {10.4007/annals.2008.167.481}, url = {https://doi.org/10.4007/annals.2008.167.481}, zblnumber = {1191.11025}, }
[Greenleaf69] F. P. Greenleaf, Invariant Means on Topological Groups and their Applications, Van Nostrand Reinhold Co., New York, 1969, vol. 16.

Show bibtex

@BOOK{Greenleaf69, author = {Greenleaf, Frederick P.}, title = {Invariant Means on Topological Groups and their Applications}, series = {Van Nostrand Math. Stud.}, volume = {16}, publisher = {Van Nostrand Reinhold Co., New York}, year = {1969}, pages = {ix+113}, mrclass = {46.80 (22.00)}, mrnumber = {0251549}, mrreviewer = {R. B. Burckel}, zblnumber = {0174.19001}, }
[Hindman79] $Go to document$ N. Hindman, "Ultrafilters and combinatorial number theory," in Number Theory, Carbondale 1979, Springer, Berlin, 1979, vol. 751, pp. 119-184.

Show bibtex

@INCOLLECTION{Hindman79, author = {Hindman, Neil}, title = {Ultrafilters and combinatorial number theory}, booktitle = {Number {T}heory, {C}arbondale 1979}, venue = {{P}roc. {S}outhern {I}llinois {C}onf., {S}outhern {I}llinois {U}niv., {C}arbondale, {I}ll., 1979}, series = {Lecture Notes in Math.}, volume = {751}, pages = {119--184}, publisher = {Springer, Berlin}, note = {(Melvyn B. Nathanson, ed.)}, year = {1979}, mrclass = {10A99 (04A20 05A99)}, mrnumber = {0564927}, mrreviewer = {P. Erd{ő}s}, zblnumber = {0416.10042}, doi = {10.1007/BFb0062697}, url = {https://doi.org/10.1007/BFb0062697}, }
[MR2893605] $Go to document$ N. Hindman and D. Strauss, Algebra in the Stone-\vCech Compactification, Theory and Applications, Walter de Gruyter & Co., Berlin, 2012.

Show bibtex

@BOOK{MR2893605, author = {Hindman, Neil and Strauss, Dona}, title = {Algebra in the {S}tone-\v{C}ech {C}ompactification, {T}heory and {A}pplications}, series = {De Gruyter Textbook}, titlenote = {second revised and extended edition [of \mr{1642231}]}, publisher = {Walter de Gruyter \& Co., Berlin}, year = {2012}, pages = {xviii+591}, isbn = {978-3-11-025623-9}, mrclass = {54-02 (03E05 22A15 54D35 54H99)}, mrnumber = {2893605}, zblnumber = {1241.22001}, doi = {10.1515/9783110258356}, url = {https://doi.org/10.1515/9783110258356}, }
[MR0677568] $Go to document$ N. Hindman, "On density, translates, and pairwise sums of integers," J. Combin. Theory Ser. A, vol. 33, iss. 2, pp. 147-157, 1982.

Show bibtex

@ARTICLE{MR0677568, author = {Hindman, Neil}, title = {On density, translates, and pairwise sums of integers}, journal = {J. Combin. Theory Ser. A}, fjournal = {Journal of Combinatorial Theory. Series A}, volume = {33}, year = {1982}, number = {2}, pages = {147--157}, issn = {0097-3165}, mrclass = {10L02}, mrnumber = {0677568}, mrreviewer = {T. C. Brown}, doi = {10.1016/0097-3165(82)90003-6}, url = {https://doi.org/10.1016/0097-3165(82)90003-6}, zblnumber = {0496.10036}, }
[Host_Kra09] $Go to document$ B. Host and B. Kra, "Uniformity seminorms on $\ell^\infty$ and applications," J. Anal. Math., vol. 108, pp. 219-276, 2009.

Show bibtex

@ARTICLE{Host_Kra09, author = {Host, Bernard and Kra, Bryna}, title = {Uniformity seminorms on {$\ell^\infty$} and applications}, journal = {J. Anal. Math.}, fjournal = {Journal d'Analyse Mathématique}, volume = {108}, year = {2009}, pages = {219--276}, issn = {0021-7670}, mrclass = {11B30 (11B25 28D05 37A45)}, mrnumber = {2544760}, mrreviewer = {Sergeĭ V. Konyagin}, doi = {10.1007/s11854-009-0024-1}, url = {https://doi.org/10.1007/s11854-009-0024-1}, zblnumber = {1183.37011}, }
[MR2755924] $Go to document$ B. Host and B. Kra, "Nil-Bohr sets of integers," Ergodic Theory Dynam. Systems, vol. 31, iss. 1, pp. 113-142, 2011.

Show bibtex

@ARTICLE{MR2755924, author = {Host, Bernard and Kra, Bryna}, title = {Nil-{B}ohr sets of integers}, journal = {Ergodic Theory Dynam. Systems}, volume = {31}, number = {1}, year = {2011}, pages = {113--142}, mrnumber = {2755924}, doi = {10.1017/S014338570900087X}, zblnumber = {1209.37007}, }
[MR0077092] $Go to document$ K. Jacobs, "Ergodentheorie und fastperiodische Funktionen auf Halbgruppen," Math. Z., vol. 64, pp. 298-338, 1956.

Show bibtex

@ARTICLE{MR0077092, author = {Jacobs, Konrad}, title = {Ergodentheorie und fastperiodische {F}unktionen auf {H}albgruppen}, journal = {Math. Z.}, fjournal = {Mathematische Zeitschrift}, volume = {64}, year = {1956}, pages = {298--338}, issn = {0025-5874}, mrclass = {46.2X}, mrnumber = {0077092}, mrreviewer = {M. M. Day}, doi = {10.1007/BF01166575}, url = {https://doi.org/10.1007/BF01166575}, zblnumber = {0070.11701}, }
[Koopman_vonNeumann32] $Go to document$ B. O. Koopman and J. v. Neumann, "Dynamical systems of continuous spectra," Proc. Natl. Acad. Sci., vol. 18, iss. 3, pp. 255-263, 1932.

Show bibtex

@ARTICLE{Koopman_vonNeumann32, author = {Koopman, B. O. and Neumann, J. v.}, title = {Dynamical systems of continuous spectra}, journal = {Proc. Natl. Acad. Sci.}, volume = {18}, number = {3}, year = {1932}, pages = {255--263}, zblnumber = {0006.22702}, doi = {10.1073/pnas.18.3.255}, url = {https://doi.org/10.1073/pnas.18.3.255}, }
[MR0797411] $Go to document$ U. Krengel, Ergodic Theorems, Walter de Gruyter & Co., Berlin, 1985, vol. 6.

Show bibtex

@BOOK{MR0797411, author = {Krengel, Ulrich}, title = {Ergodic Theorems}, series = {De Gruyter Stud. Math.}, volume = {6}, note = {with a supplement by {A}ntoine {B}runel}, publisher = {Walter de Gruyter \& Co., Berlin}, year = {1985}, pages = {viii+357}, isbn = {3-11-008478-3}, mrclass = {28-02 (28Dxx 47A35)}, mrnumber = {0797411}, mrreviewer = {E. Flytzanis}, doi = {10.1515/9783110844641}, url = {https://doi.org/10.1515/9783110844641}, zblnumber = {0575.28009}, }
[MR0131784] $Go to document$ K. de Leeuw and I. Glicksberg, "Applications of almost periodic compactifications," Acta Math., vol. 105, pp. 63-97, 1961.

Show bibtex

@ARTICLE{MR0131784, author = {de Leeuw, K. and Glicksberg, I.}, title = {Applications of almost periodic compactifications}, journal = {Acta Math.}, fjournal = {Acta Mathematica}, volume = {105}, year = {1961}, pages = {63--97}, issn = {0001-5962}, mrclass = {22.65 (42.50)}, mrnumber = {0131784}, mrreviewer = {H. Umegaki}, doi = {10.1007/BF02559535}, url = {https://doi.org/10.1007/BF02559535}, zblnumber = {0104.05501}, }
[MR2122919] $Go to document$ A. Leibman, "Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold," Ergodic Theory Dynam. Systems, vol. 25, iss. 1, pp. 201-213, 2005.

Show bibtex

@ARTICLE{MR2122919, author = {Leibman, A.}, title = {Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold}, journal = {Ergodic Theory Dynam. Systems}, fjournal = {Ergodic Theory and Dynamical Systems}, volume = {25}, year = {2005}, number = {1}, pages = {201--213}, issn = {0143-3857}, mrclass = {37A17 (22F30 28D15)}, mrnumber = {2122919}, mrreviewer = {Alexander Gorodnik}, doi = {10.1017/S0143385704000215}, url = {https://doi.org/10.1017/S0143385704000215}, zblnumber = {1080.37003}, }
[MR1865397] $Go to document$ E. Lindenstrauss, "Pointwise theorems for amenable groups," Invent. Math., vol. 146, iss. 2, pp. 259-295, 2001.

Show bibtex

@ARTICLE{MR1865397, author = {Lindenstrauss, Elon}, title = {Pointwise theorems for amenable groups}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {146}, year = {2001}, number = {2}, pages = {259--295}, issn = {0020-9910}, mrclass = {37A15 (28D15 37A30)}, mrnumber = {1865397}, mrreviewer = {Thomas Ward}, doi = {10.1007/s002220100162}, url = {https://doi.org/10.1007/s002220100162}, zblnumber = {1038.37004}, }
[Nathanson80] $Go to document$ M. B. Nathanson, "Sumsets contained in infinite sets of integers," J. Combin. Theory Ser. A, vol. 28, iss. 2, pp. 150-155, 1980.

Show bibtex

@ARTICLE{Nathanson80, author = {Nathanson, Melvyn B.}, title = {Sumsets contained in infinite sets of integers}, journal = {J. Combin. Theory Ser. A}, fjournal = {Journal of Combinatorial Theory. Series A}, volume = {28}, year = {1980}, number = {2}, pages = {150--155}, issn = {0097-3165}, mrclass = {10L02}, mrnumber = {0563552}, mrreviewer = {N. Hindman}, doi = {10.1016/0097-3165(80)90081-3}, url = {https://doi.org/10.1016/0097-3165(80)90081-3}, zblnumber = {0451.10036}, }
[MR1576401] $Go to document$ F. P. Ramsey, "On a problem of formal logic," Proc. London Math. Soc. (2), vol. 30, iss. 4, pp. 264-286, 1929.

Show bibtex

@ARTICLE{MR1576401, author = {Ramsey, F. P.}, title = {On a problem of formal logic}, journal = {Proc. London Math. Soc. (2)}, fjournal = {Proceedings of the London Mathematical Society. Second Series}, volume = {30}, year = {1929}, number = {4}, pages = {264--286}, issn = {0024-6115}, mrclass = {DML}, mrnumber = {1576401}, doi = {10.1112/plms/s2-30.1.264}, url = {https://doi.org/10.1112/plms/s2-30.1.264}, jfmnumber = {55.0032.04}, }
[MR0695335] I. Z. Ruzsa, "Uniform distribution, positive trigonometric polynomials and difference sets," in Seminar on Number Theory, 1981/1982, Univ. Bordeaux I, Talence, 1982, p. x.

Show bibtex

@INCOLLECTION{MR0695335, author = {Ruzsa, Imre Z.}, title = {Uniform distribution, positive trigonometric polynomials and difference sets}, booktitle = {Seminar on {N}umber {T}heory, 1981/1982}, pages = {Exp. No. 18, 18}, publisher = {Univ. Bordeaux I, Talence}, year = {1982}, mrclass = {10K30 (05B10 10K05)}, mrnumber = {0695335}, mrreviewer = {L. Kuipers}, zblnumber = {0515.10048}, }
[Tomkowicz_Wagon16] $Go to document$ G. Tomkowicz and S. Wagon, The Banach-Tarski Paradox, Second ed., Cambridge University Press, New York, 2016, vol. 163.

Show bibtex

@BOOK{Tomkowicz_Wagon16, author = {Tomkowicz, Grzegorz and Wagon, Stan}, title = {The {B}anach-{T}arski Paradox}, series = {Encycl. Math. Appl.}, volume = {163}, edition = {Second}, note = {With a foreword by Jan Mycielski}, publisher = {Cambridge University Press, New York}, year = {2016}, pages = {12--35}, isbn = {978-1-107-04259-9}, mrclass = {03-02 (01A75 03E15 03E25 28-02 43A05)}, mrnumber = {3616119}, mrreviewer = {Klaas Pieter Hart}, zblnumber = {1372.43001}, doi = {10.1017/CBO9781107337145.006}, url = {https://doi.org/10.1017/CBO9781107337145.006}, }
[MR648108] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982, vol. 79.

Show bibtex

@BOOK{MR648108, author = {Walters, Peter}, title = {An Introduction to Ergodic Theory}, series = {Grad. Texts in Math.}, volume = {79}, publisher = {Springer-Verlag, New York-Berlin}, year = {1982}, pages = {ix+250}, isbn = {0-387-90599-5}, mrclass = {28Dxx (54H20 58F11)}, mrnumber = {0648108}, mrreviewer = {M. A. Akcoglu}, zblnumber = {0958.28011}, }

A proof of a sumset conjecture of Erdős

Abstract

Authors