A proof of a sumset conjecture of Erdős

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.

  • [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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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.
    @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},
      }

Authors

Joel Moreira

Northwestern University, Evanston, IL

Florian K. Richter

Northwestern University, Evanston, IL

Donald Robertson

The University of Utah, Salt Lake City, UT