Let $ P:\F\times\F\to\F $ be a polynomial of bounded degree over a finite field $\F $ of large
characteristic. In this paper we establish the following dichotomy: either $ P $ is a moderate …

The\emph {sum-product phenomenon} predicts that a finite set $ A $ in a ring $ R $ should
have either a large sumset $ A+ A $ or large product set $ A\cdot A $ unless it is in some …

Abstract For E ⊂\mathbb F_q^ d, d ≥ 2, where\mathbb F_q is the finite field with q elements,
we consider the distance graph\mathcal G^ dist _t (E), t ≠ 0, where the vertices are the …

Let $ E\subset {\Bbb F} _q^ d $, the $ d $-dimensional vector space over a finite field with $ q
$ elements. Construct a graph, called the distance graph of $ E $, by letting the vertices be …

Let $ E $ be a subset in $\mathbb {F} _p^ 2$ and $ S $ be a subset in the special linear
group $ SL_2 (\mathbb {F} _p) $ or the $1 $-dimensional Heisenberg linear group $\mathbb …

We prove that for some universal c, a non-collinear set of N> 1 c points in the Euclidean
plane determines at least c N log N distinct areas of triangles with one vertex at the origin, as …

We prove three theorems giving extremal bounds on the incidence structures determined by
subsets of the points and blocks of a balanced incomplete block design (BIBD). These …

The first main result of this paper establishes that any sufficiently large subset of a plane
over the finite field F _q, namely any set E ⊆ F _q^ 2 of cardinality| E|> q, determines at least …

Abstract Given E⊂ ℝ d, define the volume set of E, 𝒱 (E)={det (x 1, x 2,..., xd): xj∈ E}. In ℝ3,
we prove that 𝒱 (E) has positive Lebesgue measure if either the Hausdorff dimension of E⊂ …

Let $\mathbb {F} _q^ d $ be the $ d $-dimensional vector space over the finite field with $ q $
elements. For a subset $ E\subseteq\mathbb {F} _q^ d $ and a fixed nonzero $ t\in\mathbb …