Abstract
We show that for an integer $n\ge 1$, any subset $A\subseteq \mathbb{Z}_4^n$ free of three-term arithmetic progressions has size $|A|\le 4^{\gamma n}$, with an absolute constant $\gamma\approx 0.926$.
-
[b:bk]
M. Bateman and N. H. Katz, "New bounds on cap sets," J. Amer. Math. Soc., vol. 25, iss. 2, pp. 585-613, 2012.
@ARTICLE{b:bk, mrkey = {2869028},
number = {2},
issn = {0894-0347},
author = {Bateman, Michael and Katz, Nets Hawk},
mrclass = {11B30 (05D40 11B75)},
doi = {10.1090/S0894-0347-2011-00725-X},
journal = {J. Amer. Math. Soc.},
zblnumber = {1262.11010},
volume = {25},
mrnumber = {2869028},
fjournal = {Journal of the American Mathematical Society},
mrreviewer = {Peter James Dukes},
title = {New bounds on cap sets},
year = {2012},
pages = {585--613},
} -
[b:bl]
T. F. Bloom, "A quantitative improvement for Roth’s theorem on arithmetic progressions," J. Lond. Math. Soc., vol. 93, iss. 3, pp. 643-663, 2016.
@ARTICLE{b:bl, mrkey = {3509957},
number = {3},
issn = {0024-6107},
author = {Bloom, T. F.},
mrclass = {11B25 (11B30 11T55)},
doi = {10.1112/jlms/jdw010},
journal = {J. Lond. Math. Soc.},
zblnumber = {06618266},
volume = {93},
mrnumber = {3509957},
fjournal = {Journal of the London Mathematical Society. Second Series},
title = {A quantitative improvement for {R}oth's theorem on arithmetic progressions},
year = {2016},
pages = {643--663},
} -
[b:b]
J. Bourgain, "On triples in arithmetic progression," Geom. Funct. Anal., vol. 9, iss. 5, pp. 968-984, 1999.
@ARTICLE{b:b, mrkey = {1726234},
number = {5},
issn = {1016-443X},
author = {Bourgain, J.},
mrclass = {11P55 (11B25 11N64)},
doi = {10.1007/s000390050105},
journal = {Geom. Funct. Anal.},
zblnumber = {0959.11004},
volume = {9},
mrnumber = {1726234},
fjournal = {Geometric and Functional Analysis},
mrreviewer = {Serge{\u\i} V. Konyagin},
coden = {GFANFB},
title = {On triples in arithmetic progression},
year = {1999},
pages = {968--984},
} -
[b:bb]
T. C. Brown and J. P. Buhler, "A density version of a geometric Ramsey theorem," J. Combin. Theory Ser. A, vol. 32, iss. 1, pp. 20-34, 1982.
@ARTICLE{b:bb, mrkey = {0640624},
number = {1},
issn = {0097-3165},
author = {Brown, T. C. and Buhler, J. P.},
mrclass = {05A17 (05C55 10L10)},
doi = {10.1016/0097-3165(82)90062-0},
journal = {J. Combin. Theory Ser. A},
zblnumber = {0476.51008},
volume = {32},
mrnumber = {0640624},
fjournal = {Journal of Combinatorial Theory. Series A},
mrreviewer = {R. L. Graham},
coden = {JCBTA7},
title = {A density version of a geometric {R}amsey theorem},
year = {1982},
pages = {20--34},
} -
[b:fgr]
P. Frankl, R. L. Graham, and V. Rödl, "On subsets of abelian groups with no $3$-term arithmetic progression," J. Combin. Theory Ser. A, vol. 45, iss. 1, pp. 157-161, 1987.
@ARTICLE{b:fgr, mrkey = {0883900},
number = {1},
issn = {0097-3165},
author = {Frankl, P. and Graham, R. L. and R{ö}dl, V.},
mrclass = {05A99 (20D60 20K01)},
doi = {10.1016/0097-3165(87)90053-7},
journal = {J. Combin. Theory Ser. A},
zblnumber = {0613.10043},
volume = {45},
mrnumber = {0883900},
fjournal = {Journal of Combinatorial Theory. Series A},
mrreviewer = {Joe P. Buhler},
coden = {JCBTA7},
title = {On subsets of abelian groups with no {$3$}-term arithmetic progression},
year = {1987},
pages = {157--161},
} -
[b:h]
D. R. Heath-Brown, "Integer sets containing no arithmetic progressions," J. London Math. Soc., vol. 35, iss. 3, pp. 385-394, 1987.
@ARTICLE{b:h, mrkey = {0889362},
number = {3},
issn = {0024-6107},
author = {Heath-Brown, D. R.},
mrclass = {11B25 (11N35)},
doi = {10.1112/jlms/s2-35.3.385},
journal = {J. London Math. Soc.},
zblnumber = {0589.10062},
volume = {35},
mrnumber = {0889362},
fjournal = {Journal of the London Mathematical Society. Second Series},
mrreviewer = {Andr{á}s S{á}rk{ö}zy},
coden = {JLMSAK},
title = {Integer sets containing no arithmetic progressions},
year = {1987},
pages = {385--394},
} -
[b:l1]
V. F. Lev, "Progression-free sets in finite abelian groups," J. Number Theory, vol. 104, iss. 1, pp. 162-169, 2004.
@ARTICLE{b:l1, mrkey = {2021632},
number = {1},
issn = {0022-314X},
author = {Lev, Vsevolod F.},
mrclass = {11B75 (20K01)},
doi = {10.1016/S0022-314X(03)00148-3},
journal = {J. Number Theory},
zblnumber = {1043.11022},
volume = {104},
mrnumber = {2021632},
fjournal = {Journal of Number Theory},
mrreviewer = {David J. Grynkiewicz},
coden = {JNUTA9},
title = {Progression-free sets in finite abelian groups},
year = {2004},
pages = {162--169},
} -
[b:l2]
V. F. Lev, "Character-free approach to progression-free sets," Finite Fields Appl., vol. 18, iss. 2, pp. 378-383, 2012.
@ARTICLE{b:l2, mrkey = {2890558},
number = {2},
issn = {1071-5797},
author = {Lev, Vsevolod F.},
mrclass = {11B25 (11B75)},
doi = {10.1016/j.ffa.2011.09.006},
journal = {Finite Fields Appl.},
zblnumber = {1284.11020},
volume = {18},
mrnumber = {2890558},
fjournal = {Finite Fields and their Applications},
mrreviewer = {Ben Joseph Green},
title = {Character-free approach to progression-free sets},
year = {2012},
pages = {378--383},
} -
[b:mcwsl] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North-Holland Publ. Co., 1977.
@BOOK{b:mcwsl,
author = {MacWilliams, F. J. and Sloane, N. J. A.},
title = {The Theory of Error-Correcting Codes},
publisher = {North-Holland Publ. Co.},
address = {Amsterdam},
year = {1977},
zblnumber = {0369.94008},
mrnumber = {0465509},
} -
[b:m]
R. Meshulam, "On subsets of finite abelian groups with no $3$-term arithmetic progressions," J. Combin. Theory Ser. A, vol. 71, iss. 1, pp. 168-172, 1995.
@ARTICLE{b:m, mrkey = {1335785},
number = {1},
issn = {0097-3165},
author = {Meshulam, Roy},
mrclass = {20D60 (05D10 11P99)},
doi = {10.1016/0097-3165(95)90024-1},
journal = {J. Combin. Theory Ser. A},
zblnumber = {0832.11006},
volume = {71},
mrnumber = {1335785},
fjournal = {Journal of Combinatorial Theory. Series A},
coden = {JCBTA7},
title = {On subsets of finite abelian groups with no {$3$}-term arithmetic progressions},
year = {1995},
pages = {168--172},
} -
[b:r1] K. Roth, "Sur quelques ensembles d’entiers," C. R. Acad. Sci. Paris, vol. 234, pp. 388-390, 1952.
@ARTICLE{b:r1, mrkey = {0046374},
author = {Roth, Klaus},
mrclass = {10.0X},
journal = {C. R. Acad. Sci. Paris},
zblnumber = {0046.04302},
volume = {234},
mrnumber = {0046374},
mrreviewer = {P. Erd{ö}s},
title = {Sur quelques ensembles d'entiers},
year = {1952},
pages = {388--390},
} -
@ARTICLE{b:r2, mrkey = {0051853},
issn = {0024-6107},
author = {Roth, Klaus},
mrclass = {10.0X},
doi = {10.1112/jlms/s1-28.1.104},
journal = {J. London Math. Soc.},
zblnumber = {0050.04002},
volume = {28},
mrnumber = {0051853},
fjournal = {Journal of the London Mathematical Society. Second Series},
mrreviewer = {P. Erd{ö}s},
title = {On certain sets of integers},
year = {1953},
pages = {104--109},
} -
@ARTICLE{b:s1, mrkey = {2560257},
number = {2},
issn = {1948-206X},
author = {Sanders, Tom},
mrclass = {11B30 (11B25)},
doi = {10.2140/apde.2009.2.211},
journal = {Anal. PDE},
zblnumber = {1197.11017},
volume = {2},
mrnumber = {2560257},
fjournal = {Analysis \& PDE},
mrreviewer = {William A. Cherry},
title = {Roth's theorem in {$\Bbb Z\sp n\sb 4$}},
year = {2009},
pages = {211--234},
} -
@ARTICLE{b:s2, mrkey = {2892617},
issn = {0021-7670},
author = {Sanders, Tom},
mrclass = {11B25 (11B05)},
doi = {10.1007/s11854-012-0003-9},
journal = {J. Anal. Math.},
zblnumber = {1280.11009},
volume = {116},
mrnumber = {2892617},
fjournal = {Journal d'Analyse Mathématique},
mrreviewer = {Mei Chu Chang},
coden = {JOAMAV},
title = {On certain other sets of integers},
year = {2012},
pages = {53--82},
} -
[b:s3]
T. Sanders, "On Roth’s theorem on progressions," Ann. of Math., vol. 174, iss. 1, pp. 619-636, 2011.
@ARTICLE{b:s3, mrkey = {2811612},
number = {1},
issn = {0003-486X},
author = {Sanders, Tom},
mrclass = {11B25 (11B30)},
doi = {10.4007/annals.2011.174.1.20},
journal = {Ann. of Math.},
zblnumber = {1264.11004},
volume = {174},
mrnumber = {2811612},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {Julia Wolf},
coden = {ANMAAH},
title = {On {R}oth's theorem on progressions},
year = {2011},
pages = {619--636},
} -
[b:sz]
E. Szemerédi, "Integer sets containing no arithmetic progressions," Acta Math. Hungar., vol. 56, iss. 1-2, pp. 155-158, 1990.
@ARTICLE{b:sz, mrkey = {1100788},
number = {1-2},
issn = {0236-5294},
author = {Szemer{é}di, E.},
mrclass = {11N64 (11B25 11P55)},
doi = {10.1007/BF01903717},
journal = {Acta Math. Hungar.},
zblnumber = {0721.11007},
volume = {56},
mrnumber = {1100788},
fjournal = {Acta Mathematica Hungarica},
mrreviewer = {D. R. Heath-Brown},
title = {Integer sets containing no arithmetic progressions},
year = {1990},
pages = {155--158},
}