We consider four related problems.(1) Obtaining dimension estimates for the set of
exceptional vantage points for the pinned Falconer distance problem.(2) Nonlinear …

Given, we call a polynomial degenerate if there exist and monomials with fractional
exponents, such that. Our main result shows that whenever a polynomial, with degree, is …

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn
Repeated Patterns in Proper Colourings, preprint, https://arxiv. org/abs/2002.00921 (2020) …

A theorem of Elekes and Szabó recognizes algebraic groups among certain complex
algebraic varieties with maximal size intersections with finite grids. We establish a …

This thesis-naturally delineated into two parts-attempts to address two flavors of problems in
arithmetic combinatorics. These two parts are based on the papers\cite {JRT} and\cite {BFR} …

In this dissertation we study arithmetic Ramsey type problems and inverse problems, in
various settings. This work consists of two parts. In Part I, we study arithmetic Ramsey type …

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn
(arXiv: 2002.00921). Given a graph H and an integer k⩾ 2, let fk (n, H) be the smallest …