New results on sum-product type growth over fields

TL;DR: In this paper, the authors obtained lower bounds on the size of finite subsets of a field in the regime when the sizes of the subsets are different significantly from each other.

TL;DR: For a general class of 3-point configurations, the configuration set of a set of sets has non-empty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing the Sobolev estimates of associated generalized Radon transforms as discussed by the authors.

TL;DR: In this paper, it was shown that the set of nonzero values of a skew-symmetric bilinear form of a point set has cardinality π(N −9/13).

TL;DR: The current best bound of 4/3+5/5277 is due to Shakan et al. as discussed by the authors, who showed that the product set of the difference set of a polynomial subset of a finite field can have size at least 2 −1 −1/13542+o(1) for a small positive constant.

TL;DR: Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.

TL;DR: In this paper, it was shown that a set of points in R 2 has at least c N log N distinct distances, thus obtaining the sharp exponent in a problem of Erd} os.

TL;DR: In this paper, the Szemeredi-Trotter type theorem for finite fields was shown to be equivalent to the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem.

TL;DR: In this paper, the integers of the form ======¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯󾾷�ﾷ (1) were considered, where n is a sequence of integers, and n is the number of nodes in the sequence.