Geodesic
Bounds for sets with no polynomial progressions
Forum of Mathematics, Pi, Tome 8 (2020)

Voir la notice de l'article provenant de la source Cambridge University Press

Let $P_1,\dots ,P_m\in \mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of $\{1,\dots ,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots ,x+P_m(y)$ has size $|A|\ll N/(\log \log {N})^{c_{P_1,\dots ,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms. 
@article{10_1017_fmp_2020_11,
     author = {Sarah Peluse},
     title = {Bounds for sets with no polynomial progressions},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {8},
     year = {2020},
     doi = {10.1017/fmp.2020.11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2020.11/}
}
TY  - JOUR
AU  - Sarah Peluse
TI  - Bounds for sets with no polynomial progressions
JO  - Forum of Mathematics, Pi
PY  - 2020
VL  - 8
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2020.11/
DO  - 10.1017/fmp.2020.11
LA  - en
ID  - 10_1017_fmp_2020_11
ER  - 
%0 Journal Article
%A Sarah Peluse
%T Bounds for sets with no polynomial progressions
%J Forum of Mathematics, Pi
%D 2020
%V 8
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2020.11/
%R 10.1017/fmp.2020.11
%G en
%F 10_1017_fmp_2020_11
Sarah Peluse. Bounds for sets with no polynomial progressions. Forum of Mathematics, Pi, Tome 8 (2020). doi: 10.1017/fmp.2020.11

[1] Balog, A., Pelikán, J., Pintz, J. and Szemerédi, E., ‘Difference sets without th powers’, Acta Math. Hungar. 65(2) (1994), 165–187.10.1007/BF01874311Google Scholar | DOI

[2] Bergelson, V. and Leibman, A., ‘Polynomial extensions of van der Waerden’s and Szemerédi’s theorems’, J. Amer. Math. Soc. 9(3) (1996), 725–753.10.1090/S0894-0347-96-00194-4Google Scholar | DOI

[3] Bloom, T. F., ‘A quantitative improvement for Roth’s theorem on arithmetic progressions’, J. Lond. Math. Soc. (2) 93(3) (2016), 643–663.10.1112/jlms/jdw010Google Scholar | DOI

[4] Bourgain, J. and Chang, M.-C., ‘Nonlinear Roth type theorems in finite fields’, Israel J. Math., (2017), 853–867.10.1007/s11856-017-1577-9Google Scholar | DOI

[5] Dong, D., Li, X. and Sawin, W., ‘Improved estimates for polynomial Roth type theorems in finite fields’, Preprint, 2017, .Google Scholar | arXiv

[6] Gowers, W. T., ‘A new proof of Szemerédi’s theorem for arithmetic progressions of length four’, Geom. Funct. Anal. 8(3) (1998), 529–551.10.1007/s000390050065Google Scholar | DOI

[7] Gowers, W. T., ‘A new proof of Szemerédi’s theorem’, Geom. Funct. Anal. 11(3) (2001), 465–588.10.1007/s00039-001-0332-9Google Scholar | DOI

[8] Gowers, W. T., ‘Arithmetic progressions in sparse sets’, in Current Developments in Mathematics, 2000, pp. 149–196 (International Press, Somerville, MA, 2001).Google Scholar

[9] Green, B. and Tao, T., ‘Linear equations in primes’, Ann. of Math. (2), 171(3) (2010), 1753–1850.10.4007/annals.2010.171.1753Google Scholar | DOI

[10] Green, B. and Tao, T., ‘New bounds for Szemerédi’s theorem, III: a polylogarithmic bound for ’, Mathematika 63(3) (2017), 944–1040.10.1112/S0025579317000316Google Scholar | DOI

[11] Lucier, J., ‘Intersective sets given by a polynomial’, Acta Arith. 123(1) (2006), 57–95.10.4064/aa123-1-4Google Scholar | DOI

[12] Montgomery, H. L., Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, Vol. 84 of CBMS Regional Conference Series in Mathematics (American Mathematical Society, Providence, RI, 1994).Google Scholar

[13] Peluse, S., ‘Three-term polynomial progressions in subsets of finite fields’, Israel J. Math. 228(1) (2018), 379–405.10.1007/s11856-018-1768-zGoogle Scholar | DOI

[14] Peluse, S., ‘On the polynomial Szemerédi theorem in finite fields’, Duke Math. J. 168(5) (2019), 749–774.10.1215/00127094-2018-0051Google Scholar | DOI

[15] Peluse, S. and Prendiville, S., ‘Quantitative bounds in the non-linear Roth theorem’, Preprint, 2019, .Google Scholar | arXiv

[16] Prendiville, S., ‘Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case’, Discrete Anal. 5 (2017), 34 pages.Google Scholar

[17] Rice, A., ‘A maximal extension of the best-known bounds for the Furstenberg-Sárközy theorem’, Acta Arith. 187(1) (2019), 1–41.10.4064/aa170828-26-8Google Scholar | DOI

[18] Sárközy, A., ‘On difference sets of sequences of integers. I’, Acta Math. Acad. Sci. Hungar. 31(1–2) (1978), 125–149.10.1007/BF01896079Google Scholar | DOI

[19] Sárközy, A., ‘On difference sets of sequences of integers. III’, Acta Math. Acad. Sci. Hungar. 31 (1978), 355–386.10.1007/BF01901984Google Scholar | DOI

[20] Slijepčević, S., ‘A polynomial Sárközy-Furstenberg theorem with upper bounds’, Acta Math. Hungar. 98(1–2) (2003), 111–128.10.1023/A:1022813623110Google Scholar | DOI

[21] Szemerédi, E., ‘On sets of integers containing no elements in arithmetic progression’, Acta Arith. 27 (1975), 199–245. Collection of articles in memory of Juriĭ Vladimirovič Linnik.10.4064/aa-27-1-199-245Google Scholar | DOI

[22] Tao, T., Higher Order Fourier Analysis, Vol. 142 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2012).Google Scholar

[23] Tao, T. and Ziegler, T., ‘The primes contain arbitrarily long polynomial progressions’, Acta Math. 201(2) (2008), 213–305.10.1007/s11511-008-0032-5Google Scholar | DOI

[24] Tao, T. and Ziegler, T., ‘Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors’, Discrete Anal. 60 (2016), 61 pages.Google Scholar

[25] Tao, T. and Ziegler, T., ‘Polynomial patterns in the primes’, Forum Math. Pi 6 (2018), e1, 60 pages.10.1017/fmp.2017.3Google Scholar | DOI

Cité par Sources :