Geodesic
On discrete values of bilinear forms
Sbornik. Mathematics, Tome 209 (2018) no. 10, pp. 1482-1497
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $\omega$ be a nondegenerate skew-symmetric bilinear form in the real plane. We prove that for finite a point set $P\subset \mathbb R^2\setminus\{0\}$, the set $T_\omega(P)$ of nonzero values of $\omega$ on $P\times P$, if nonempty, has cardinality $\Omega(N^{96/137})$. In the special case when $P=A\times A$, where $A$ is a set of at least two reals, we establish the following sum-product type estimates, corresponding to the symmetric and skew-symmetric form $\omega$: $$ |AA+ AA|= \Omega(|A|^{19/12}) \quad\text{and}\quad |AA-AA|= \Omega\biggl( \frac{|A|^{49/32}}{\log^{3/32}|A|}\biggr). $$ These estimates improve their basic prototypes $\Omega(N^{2/3})$ and $\Omega(|A|^{3/2})$, which readily follow from the Szemerédi-Trotter theorem. Bibliography: 28 titles.
Keywords: Erdős problems, sum-product estimates, cross-ratio. 
@article{SM_2018_209_10_a3,
     author = {A. Iosevich and O. Roche-Newton and M. Rudnev},
     title = {On discrete values of bilinear forms},
     journal = {Sbornik. Mathematics},
     pages = {1482--1497},
     year = {2018},
     volume = {209},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_10_a3/}
}
TY  - JOUR
AU  - A. Iosevich
AU  - O. Roche-Newton
AU  - M. Rudnev
TI  - On discrete values of bilinear forms
JO  - Sbornik. Mathematics
PY  - 2018
SP  - 1482
EP  - 1497
VL  - 209
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2018_209_10_a3/
LA  - en
ID  - SM_2018_209_10_a3
ER  - 
%0 Journal Article
%A A. Iosevich
%A O. Roche-Newton
%A M. Rudnev
%T On discrete values of bilinear forms
%J Sbornik. Mathematics
%D 2018
%P 1482-1497
%V 209
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2018_209_10_a3/
%G en
%F SM_2018_209_10_a3
A. Iosevich; O. Roche-Newton; M. Rudnev. On discrete values of bilinear forms. Sbornik. Mathematics, Tome 209 (2018) no. 10, pp. 1482-1497. http://geodesic.mathdoc.fr/item/SM_2018_209_10_a3/

[1] A. Balog, “A note on sum-product estimates”, Publ. Math. Debrecen, 79:3-4 (2011), 283–289 | DOI | MR | Zbl

[2] J. Bourgain, Mei-Chu Chang, “On the size of $k$-fold sum and product sets of integers”, J. Amer. Math. Soc., 17:2 (2004), 473–497 | DOI | MR | Zbl

[3] P. Brass, W. O. J. Moser, J. Pach, Research problems in discrete geometry, Springer, New York, 2005, xii+499 pp. | DOI | MR | Zbl

[4] Mei-Chu Chang, “Sum and product of different sets”, Contrib. Discrete Math., 1:1 (2006), 47–56 | MR | Zbl

[5] L. Guth, N. H. Katz, “On the Erdős distinct distances problem in the plane”, Ann. of Math. (2), 181:1 (2015), 155–190 | DOI | MR | Zbl

[6] G. Elekes, M. Sharir, “Incidences in three dimensions and distinct distances in the plane”, Combin. Probab. Comput., 20:4 (2011), 571–608 | DOI | MR | Zbl

[7] P. Erdős, E. Szemerédi, “On sums and products of integers”, Studies in pure mathematics, Birkhäuser, Basel, 1983, 213–218 | MR | Zbl

[8] A. Iosevich, S. Konyagin, M. Rudnev, V. Ten, “Combinatorial complexity of convex sequences”, Discrete Comput. Geom., 35:1 (2006), 143–158 | DOI | MR | Zbl

[9] A. Iosevich, O. Roche-Newton, M. Rudnev, “On an application of Guth–Katz theorem”, Math. Res. Lett., 18:4 (2011), 691–697 | DOI | MR | Zbl

[10] S. V. Konyagin, M. Rudnev, “On new sum-product-type estimates”, SIAM J. Discrete Math., 27:2 (2013), 973–990 | DOI | MR | Zbl

[11] S. V. Konyagin, I. D. Shkredov, “On sum sets of sets having small product set”, Proc. Steklov Inst. Math., 290:1 (2015), 288–299 | DOI | DOI | MR | Zbl

[12] S. V. Konyagin, I. D. Shkredov, “New results on sums and products in $\mathbb R$”, Proc. Steklov Inst. Math., 294 (2016), 78–88 | DOI | DOI | MR | Zbl

[13] Liangpan Li, O. Roche-Newton, “Convexity and a sum-product type estimate”, Acta Arith., 156:3 (2012), 247–255 | DOI | MR | Zbl

[14] B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev, I. D. Shkredov, New results on sum-product type growth over fields, arXiv: 1702.01003v3

[15] J. Pach, G. Tardos, “Isosceles triangles determined by a planar point set”, Graph theory and discrete geometry (Manila, 2001), Graphs Combin., 18, no. 4, 2002, 769–779 | DOI | MR | Zbl

[16] O. Roche-Newton, M. Rudnev, “On the Minkowski distances and products of sum sets”, Israel J. Math., 209:2 (2015), 507–526 | DOI | MR | Zbl

[17] O. Roche-Newton, M. Rudnev, I. D. Shkredov, “New sum-product type estimates over finite fields”, Adv. Math., 293 (2016), 589–605 | DOI | MR | Zbl

[18] M. Rudnev, “On distinct cross-ratios and related growth problems”, Mosc. J. Comb. Number Theory, 7:3 (2017), 51–65 | MR

[19] M. Rudnev, “On the number of incidences between points and planes in three dimensions”, Combinatorica, 38:1 (2018), 219–254 | DOI | MR | Zbl

[20] M. Rudnev, J. M. Selig, “On the use of the Klein quadric for geometric incidence problems in two dimensions”, SIAM J. Discrete Math., 30:2 (2016), 934–954 | DOI | MR | Zbl

[21] T. Schoen, I. D. Shkredov, “Higher moments of convolutions”, J. Number Theory, 133:5 (2013), 1693–1737 | DOI | MR | Zbl

[22] I. D. Shkredov, “On a question of A. Balog”, Pacific J. Math., 280:1 (2016), 227–240 | DOI | MR | Zbl

[23] I. D. Shkredov, “Some remarks on sets with small quotient set”, Sb. Math., 208:12 (2017), 1854–1868 | DOI | DOI | MR | Zbl

[24] J. Solymosi, “Bounding multiplicative energy by the sumset”, Adv. Math., 222:2 (2009), 402–408 | DOI | MR | Zbl

[25] J. Solymosi, G. Tardos, “On the number of $k$-rich transformations”, Computational geometry (SCG' 07), ACM, New York, 2007, 227–231 | DOI | MR | Zbl

[26] E. Szemerédi, W. T. Trotter, Jr., “Extremal problems in discrete geometry”, Combinatorica, 3:3-4 (1983), 381–392 | DOI | MR | Zbl

[27] T. Tao, Van H. Vu, Additive combinatorics, Cambridge Stud. Adv. Math., 105, Cambridge Univ. Press, Cambridge, 2006, xviii+512 pp. | DOI | MR | Zbl

[28] C. D. Tóth, “The Szemerédi–Trotter theorem in the complex plane”, Combinatorica, 35:1 (2015), 95–126 | DOI | MR | Zbl