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In this paper, we introduce a new and direct approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let $B (\cdot, \cdot)$ be a non-degenerate bilinear form and $E$ be a set in $\mathbb{F}_q^2$. We prove that if $|E|\gg q^{5/3}$ then the number of triples $(B(x, y), B(y, z), B(z, x))$ with $x, y, z\in E$ is at least $cq^3$ for some positive constant $c$. This significantly improves a result due to the fifth listed author (2009)., Comment: 12 pages

We give an upper bound on the number of exceptional orthogonal projections of a small set of points in a plane over a prime order field, which makes progress toward a conjecture of Chen (2018). This theorem relies on a new upper bound on the number of incidences between an arbitrary set of lines and a set of points contained in a reasonably small pencil of lines. We also prove a new upper bound on the number of incidences between points and lines when both are Cartesian products in an appropriate sense.

In this paper, we study a point-hyper plane incidence theorem in matrix rings, which generalizes all previous works in literature of this direction., Comment: 9 pages, update for general cases

In this paper, we study dot-product sets and $k$-simplices in vector spaces over finite rings. We show that if $E$ is sufficiently large then the dot-product set of $E$ covers the whole ring. In higher dimensional cases, if $E$ is sufficiently large then the set of simplices and the set of dot-product simplices determined by $E$, up to congurence, have positive densities., Comment: 28 pages

Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper, we prove that for any quadratic polynomial $f(x,y,z) \in \mathcal{R}[x,y,z]$ that is of the form $axy+R(x)+S(y)+T(z)$ for some one-variable polynomials $R, S , T$, we have \[ |f(A,B,C)| \gg \min\left\{ q^r, \frac{|A||B||C|}{q^{2r-1}}\right\}\] for any $A, B, C \subset \mathcal{R}$. We also study the sum-product type problems over finite valuation ring $\mathcal{R}.$ More precisely, we show that for any $A \subset \mathcal{R}$ with $|A| \gg q^{r-1/3}$ then $$\max\{ |A \cdot A|, |A^d + A^d|\},\max\{ |A + A|, |A^2 + A^2|\},\max\{|A-A|,|AA+AA|\} \gg |A|^{2/3}q^{r/3},$$ and $|f(A) + A| \gg |A|^{2/3}q^{r/3}$ for any one variable quadratic polynomial $f$., Comment: Theorem 1.6 is cited because it has already proved

In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x+yz, x(y+z) are moderate expanders over the matrix ring, and xy + z + t is strong expander over the matrix rings. These results generalize recent results of Y.D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author., Comment: Available Online in Journal of Number Theory, 21 pages

In this paper, we study growth rate of product of sets in the Heisenberg group over finite fields and the complex numbers. More precisely, we will give improvements and extensions of recent results due to Hegyv\'{a}ri and Hennecart (2018)., Comment: 18 pages

Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $\Delta(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|\mathcal{E}|\ge 4q^{\frac{d+1}{2}}$, then $\Delta(\mathcal{E})=\mathbb{F}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if $\mathcal{E}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $(d+1)/2$, and still cover all distances. We also show that the number of pairs of points in $\mathcal{E}$ of any given distance is close to its expected value., Comment: Final version!

The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide a new $L^2\to L^r$ extension estimate for paraboloids in dimensions $d=4k+3$ and $q\equiv 3\mod 4$, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using \textit{the first association scheme graph} to analyze energy sets, and as a consequence, we obtain new $L^p\to L^4$ extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward $L^p\to L^4$ which has stood for more than ten years. Most significantly, it follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the $L^p\to L^4$ estimates for spheres with primitive radii are much stronger than those for paraboloids. Based on new estimates, we will also clarify conjectures on finite field extension problem for spheres. This results in a reasonably complete description of finite field extension theorems for spheres. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the Erd\H{o}s-Falconer distance conjecture over finite fields. The last is to prove that the Erd\H{o}s-Falconer distance conjecture holds in odd-dimensional spaces when we study distances between two sets: one set lies on a variety (paraboloids or spheres), and the other set is arbitrary in $\mathbb{F}_q^d$., Comment: 50 pages. Final version with several improvements

In this paper we obtain a new lower bound on the Erd\H{o}s distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |\Delta(A)| \gg |A|^{\frac{1}{2}+\frac{149}{4214}}.$$ Our result gives a new lower bound of $|\Delta{(A)}|$ in the range $|A|\le p^{1+\frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower bound on the number of distinct distances between a line and a set in $\mathbb{F}_p^2$. The latter is the new feature that allows us to improve the previous bound due Stevens and de Zeeuw., Comment: V2: some typos have been fixed

In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that If $A$ is a set of $M_2(\mathbb{F}_q)$ and $|A|\gg q^{7/2}$, then we have \[|A(A+A)|, ~|A+AA|\gg q^4.\] If $A$ is a set of $SL_2(\mathbb{F}_q)$ and $|A|\gg q^{5/2}$, then we have \[|A(A+A)|, ~|A+AA|\gg q^4.\] We also obtain similar results for the cases of $A(B+C)$ and $A+BC$, where $A, B, C$ are sets in $M_2(\mathbb{F}_q)$., Comment: 31 pages

In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(\mathbb{F}_p)$ with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyv\'{a}ri and Hennecart., Comment: v2: we have added some improvements

Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough., Comment: Submitted for publication

Hegyv\'ari and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $B\cdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups., Comment: This manuscript has been updated to include referee corrections. To appear in Journal of Number Theory

We determine which quadratic polynomials in three variables are expanders over an arbitrary field $\mathbb{F}$. More precisely, we prove that for a quadratic polynomial $f\in \mathbb{F}[x,y,z]$, which is not of the form $g(h(x)+k(y)+l(z))$, we have $|f(A\times B\times C)|\gg N^{3/2}$ for any sets $A,B,C\subset \mathbb{F}$ with $|A|=|B|=|C|=N$, with $N$ not too large compared to the characteristic of $\mathbb{F}$. We give several applications. We use this result for $f=(x-y)^2+z$ to obtain new lower bounds on $|A+A^2|$ and $\max\{|A+A|,|A^2+A^2|\}$, and to prove that a Cartesian product $A\times\cdots \times A\subset \mathbb{F}^d$ determines almost $|A|^2$ distinct distances if $|A|$ is not too large., Comment: v2: Various corrections. v3: We have added bounds on |A+A^2| and |A^2+A^2

In this short note, we study the distribution of spreads in a point set $\mathcal{P} \subseteq \mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $\varepsilon > 0$, if $|\mathcal{P}| \geq (1+\varepsilon) q^{\lceil d/2 \rceil}$, then $\mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $\mathcal{P} \subset \mathbb{F}_q^d$ of size $|\mathcal{P}| = q^{\lceil d/2 \rceil}$ that determine at most one spread.

In this short note, we give a lower bound on the number of congruence classes of triangles in a small set of points in $\mathbb{F}_p^2$. More precisely, for $\mathcal{A}\subset \mathbb{F}_p^2$ with $|\mathcal{A}|\le p^{2/3}$, we prove that the number of congruence classes of triangles determined by points in $\mathcal{A}\times \mathcal{A}$ is at least $|\mathcal{A}|^{7/2}$. This note is not intended for journal publication.

In this paper we introduce a unified approach to deal with incidence problems between points and varieties over finite fields. More precisely, we prove that the number of incidences $I(\mathcal{P}, \mathcal{V})$ between a set $\mathcal{P}$ of points and a set $\mathcal{V}$ of varieties of a certain form satisfies $$\left\vert I(\mathcal{P},\mathcal{V})-\frac{|\mathcal{P}||\mathcal{V}|}{q^k}\right\vert\le q^{dk/2}\sqrt{|\mathcal{P}||\mathcal{V}|}.$$ This result is a generalization of the results of Vinh (2011), Bennett et al. (2014), and Cilleruelo et al. (2015). As applications of our incidence bounds, we obtain results on the pinned value problem and the Beck type theorem for points and spheres. Using the approach introduced, we also obtain a result on the number of distinct distances between points and lines in $\mathbb{F}_q^2$, which is the finite field analogous of a recent result of Sharir et al. (2015).

Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper we generalize and improve several well-known results, which were studied over finite fields $\mathbb{F}_q$ and finite cyclic rings $\mathbb{Z}/p^r\mathbb{Z}$, in the setting of finite valuation rings., Comment: arXiv admin note: text overlap with arXiv:1510.03479

We use methods from spectral graph theory to obtain bounds on the number of incidences between $k$-planes and $h$-planes in $\mathbb{F}_q^d$ which generalize a recent result given by Bennett, Iosevich, and Pakianathan (2014). More precisely, we prove that the number of incidences between a set $\mathcal{P}$ of $k$-planes and a set $\mathcal{H}$ of $h$-planes with $h\ge 2k+1$, which is denoted by $I(\mathcal{P},\mathcal{H})$, satisfies \[\left\vert I(\mathcal{P},\mathcal{H})-\frac{|\mathcal{P}||\mathcal{H}|}{q^{(d-h)(k+1)}}\right\vert \lesssim q^{\frac{(d-h)h+k(2h-d-k+1)}{2}}\sqrt{|\mathcal{P}||\mathcal{H}|}. \]

In this paper, we use methods from spectral graph theory to obtain some results on the sum-product problem over finite valuation rings $\mathcal{R}$ of order $q^r$ which generalize recent results given by Hegyv\'ari and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$, if $A$ and $B$ are two sets in $\mathcal{R}^*$ with $|A|=|B|=q^\alpha$, then \[\max\left\lbrace |f(A, B)|, |g(A, B)| \right\rbrace\gtrsim |A|^{1+\Delta(\alpha)},\] for some $\Delta(\alpha)>0$., Comment: To appear in European Journal of Combinatorics

We prove a Szemer\'edi-Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and S\'ark\"ozy on the solvability of the equations $a + b = cd$ and $ab + 1 = cd$ over a finite field. Other analogous results that are known to hold in finite fields are generalized to finite quasifields., Comment: Referee suggestions included. To appear in Journal of Combinatorial Theory, Series A

Let $\mathbb{F}_q$ be a finite field of $q$ elements where $q$ is a large odd prime power and $Q =a_1 x_1^{c_1}+...+a_dx_d^{c_d}\in \mathbb{F}_q[x_1,...,x_d]$, where $2\le c_i\le N$, $\gcd(c_i,q)=1$, and $a_i\in \mathbb{F}_q$ for all $1\le i\le d$. A $Q$-sphere is a set of the form $\lbrace x\in \mathbb{F}_q^d | Q(x-b)=r\rbrace$, where $b\in \mathbb{F}_q^d, r\in \mathbb{F}_q$. We prove bounds on the number of incidences between a point set $\mathcal{P}$ and a $Q$-sphere set $\mathcal{S}$, denoted by $I(\mathcal{P},\mathcal{S})$, as the following. $$| I(\mathcal{P},\mathcal{S})-\frac{|\mathcal{P}||\mathcal{S}|}{q}|\le q^{d/2}\sqrt{|\mathcal{P}||\mathcal{S}|}.$$ We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings $\mathbb{Z}_q$ where $q$ is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections $4$ and $5$, we prove a bound on the number of incidences between a random point set and a random $Q$-sphere set in $\mathbb{F}_q^d$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances., Comment: to appear in Forum Math

In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems of norm, bilinear or quadratic equations over finite fields are solvable in any large subset of vector spaces over finite fields., Comment: 28 pages

We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence., Comment: FPSAC 2009

We show that if $\mathcal{E}$ is a subset of the $d$-dimensional vector space over a finite field $\mathbbm{F}_q$ ($d \geq 3$) of cardinality $|\mathcal{E}| \geq (d-1)q^{d - 1}$, then the set of volumes of $d$-dimensional parallelepipeds determined by $\mathcal{E}$ covers $\mathbbm{F}_q$. This bound is sharp up to a factor of $(d-1)$ as taking $\mathcal{E}$ to be a $(d - 1)$-hyperplane through the origin shows.

For a prime power $q$, we study the distribution of determinent of matrices with restricted entries over a finite field $\mathbbm{F}_q$ of $q$ elements. More precisely, let $N_d (\mathcal{A}; t)$ be the number of $d \times d$ matrices with entries in $\mathcal{A}$ having determinant $t$. We show that \[ N_d (\mathcal{A}; t) = (1 + o (1)) \frac{|\mathcal{A}|^{d^2}}{q}, \] if $|\mathcal{A}| = \omega(q^{\frac{d}{2d-1}})$, $d\geqslant 4$. When $q$ is a prime and $\mathcal{A}$ is a symmetric interval $[-H,H]$, we get the same result for $d\geqslant 3$. This improves a result of Ahmadi and Shparlinski (2007)., Comment: Journal of Combinatorics and Number Theory (to appear)

A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A, B$ of vertex set $V$ of the graph such that $A \cap B = \emptyset$ and $|A| + |B| = n$, there is a vertex $z$ not in $A \cup B$ joined to each vertex of $A$ and no vertex of $B$. Few explicit families of $n$-e.c. are known for $n > 2$. In this short note, we give a new construction of 3-e.c. graphs using the notion of quadrance in the finite Euclidean space $\mathbbm{Z}_p^d$.

Given $k$ sets $\mathcal{A}_i \subseteq \mathbb{F}_q^d$ and a non-degenerate bilinear form $B$ in $\mathbb{F}_q^d$. We consider the system of $l \leq \binom{k}{2}$ bilinear equations \[ B (\tmmathbf{a}_i, \tmmathbf{a}_j) = \lambda_{i j}, \tmmathbf{a}_i \in \mathcal{A}_i, i = 1, ..., k. \] We show that the system is solvable for any $\lambda_{i j} \in \mathbb{F}_q^{*}$, $1 \leq i,j \leq k$, given that the restricted sets $\mathcal{A}_i$'s are sufficiently large., Comment: to appear in Proceedings of the American Mathematical Society

art, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erd\"os-Falconer distance conjecture holds for subsets of the unit sphere in $\mathbbm{F}_q^d$. In this note, we give a graph theoretic proof of this result.

We study a finite analog of a problem of Erdos, Hickerson and Pach on the maximum number of occurrences of a fixed angle among n directions in three-dimensional spaces.

We study a finite analog of a conjecture of Erd\"os on the sum of the squared multiplicities of the distances determined by an $n$-element point set. Our result is based on an estimate of the number of hinges in spectral graphs.

Using Fourier analysis, Covert, Hart, Iosevich and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is >= rq^2, with q^{-1/2} << r <= 1 then it contains an isometric copy of >= crq^3 triangles. In this note, we give a graph theoretic proof of this result.

A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. Chvatal conjectured that there exists an absolute constant t_0 so that every t_0-tough graph is pancyclic. This conjecture was disproved by Bauer, van den Heuvel and Schmeichel by constructing a t_0-tough triangle-free graph for every real t_0. For each finite field F_q with q odd, we consider graphs associated to the finite Euclidean plane and the finite upper half plane over F_q. These graphs have received serious attention as they have been shown to be Ramanujan (or asymptotically Ramanujan) for large q. We will show that for infinitely many q, these graphs provide further counterexamples to Chvatal's conjecture. They also provide a good constructive lower bound for the Ramsey number R(3,k).

Iosevich and Senger (2008) showed that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result.

In this paper we study the kaleidoscopic pseudo-randomness of finite Euclidean graphs using probabilistic methods. Roughly speaking, we show that sufficiently large subsets of d-dimensional vector spaces over finite fields contain every possible finite configurations.

We consider unitary graphs attached to Z_d^n using an analogue of the Euclidean distance. These graphs are shown to be integral when n is odd or the dimension d is even.

We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdos distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs R(3,k).

The quadrance between two points $A_1 = (x_1, y_1)$ and $A_2 = (x_2, y_2)$ is the number $Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2$. Let $q$ be an odd prime power and $F_q$ be the finite field with $q$ elements. The unit-quadrance graph $D_q$ has the vertex set $F_q^2$, and $X, Y \in F_q^2$ are adjacent if and only if $Q (A_1, A_2) = 1$. In this paper, we study some colouring problems for the unit-quadrance graph $D_q$ and discuss some open problems.

A divisor graph $G$ is an ordered pair $(V, E)$ where $V \subset \mathbbm{Z}$ and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph. In this note, we will prove that for any $n \geqslant 1$ and $0 \leqslant m \leqslant \binom{n}{2}$ then there exists a divisor graph of order $n$ and size $m$. We also present a simple proof of the characterization of divisor graphs which is due to Chartran, Muntean, Saenpholpant and Zhang., Comment: AWOCA 2006