Abstract
Bourgain and Chang recently showed that any subset of \(\mathbb{F}_p\) of density ≫p−1/15 contains a nontrivial progression x, x + y, x + y2. We answer a question of theirs by proving that if P1, P2 ∈ ℤ[y] are linearly independent and satisfy P1(0) = P2(0) = 0, then any subset of \(\mathbb{F}_p\) of density ≫P1,P2p−1/24 contains a nontrivial polynomial progression x, x + P1(y), x + P2(y).
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References
A. Balog, J. Pelikán, J. Pintz and E. Szemerédi, Difference sets without κth powers, Acta Mathematica Hungarica 65 (1994), 165–187.
V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemer édi’s theorems, Journal of the American Mathematical Society 9 (1996), 725–753.
J. Bourgain and M.-C. Chang, Nonlinear Roth type theorems in finite fields, Israel Journal of Mathematics 221 (2017), 853–867.
B. Bukh and J. Tsimerman, Sum-product estimates for rational functions, Proceedings of the London Mathematical Society 104 (2012), 1–26.
D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Springer, New York, 2007.
W. T. Gowers, A new proof of Szemerédi’s theorem, Geometric and Functional Analysis 11 (2001), 465–588.
D. Hart, A. Iosevich and J. Solymosi, Sum-product estimates in finite fields via Kloosterman sums, International Mathematics Research Notices (2007), no. 5, Art. ID rnm007, 14.
D. Hart, L. Li and C.-Y. Shen, Fourier analysis and expanding phenomena in finite fields, Proceedings of the American Mathematical Society 141 (2013), 461–473.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York–Heidelberg, 1977.
G. Kemper, A course in Commutative Algebra, Graduate Texts in Mathematics, Vol. 256, Springer, Heidelberg, 2011.
E. Kowalski, Exponential sums over definable subsets of finite fields, Israel Journal of Mathematics 160 (2007), 219–251.
S. Lang and A. Weil, Number of points of varieties in finite fields, American Journal of Mathematics 76 (1954), 819–827.
J. Lucier, Intersective sets given by a polynomial, Acta Arithmetica 123 (2006), 57–95.
S. Prendiville, Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case, Discrete Analysis (2017), paper no. 5.
A. Sárközy, On difference sets of sequences of integers. I, Acta Mathematica Academiae Scientiarum Hungaricae 31 (1978), 125–149.
A. Sárközy, On difference sets of sequences of integers. III, Acta Mathematica Academiae Scientiarum Hungaricae 31 (1978), 355–386.
I. D. Shkredov, On monochromatic solutions of some nonlinear equations in Z/pZ, Matematicheskie Zametki 88 (2010), 625–634.
S. Slijepčević, A polynomial Sárközy–Furstenberg theorem with upper bounds, Acta Mathematica Hungarica 98 (2003), 111–128.
T. Tao, Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets, Contributions to Discrete Mathematics 10 (2015), 22–98.
V. H. Vu, Sum-product estimates via directed expanders, Mathematical Research Letters 15 (2008), 375–388.
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Peluse, S. Three-term polynomial progressions in subsets of finite fields. Isr. J. Math. 228, 379–405 (2018). https://doi.org/10.1007/s11856-018-1768-z
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DOI: https://doi.org/10.1007/s11856-018-1768-z