Improved Bounds on Sarkozy's Theorem for Quadratic Polynomials
We extend the best known bound on the largest subset of {1,2,...,N} with no square differences to the largest possible class of quadratic polynomials.
math.CA↗
arXiv subjects
Mariah Hamel
Publications and source records attributed to Mariah Hamel.
On Sums of Sets of Primes with Positive Relative Density
In this paper we show that if $A$ is a subset of the primes with positive relative density $δ$, then $A+A$ must have positive upper density $C_1δe^{-C_2(\log(1/δ))^{2/3}(\log\log(1/δ))^{1/3}}$ in $\mathbb{N}$. Our argument applies the techniques developed by Green and Green-Tao used to find arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo $M$.
math.NT↗
Arithmetic Structure in Sparse Difference Sets
Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration $\{v_1,...,v_\ell\}$ of vectors in $\mathbb{Z}^d$, we show that if $A\subset[1,N]^d$ with $|A|/N^d\geq C N^{-1/\ell}$, then there necessarily exists $r\ne0$ such that $\{rv_1, ...,rv_\ell\}\subseteq A-A$.
math.NT↗
Sum-product inequalities with perturbation
Suppose that A is a set of n real numbers, each at least 1 apart. Define the ``perturbed sum and product sets'' S and P to be the sums a + b + f(a,b) and products (a+g(a,b))(b+h(a,b)), where f, g, and h satisfy certain upper bounds in terms of the n, |a| and |b|. We develop almost best possible lower bounds on |P| + |S|, using the largest possible sizes of the ``perturbation parameters'' f(a,b), g(a,b) and h(a,b). Our proof uses Elekes's idea for bounding |A+A|+|A.A| from below, in combination with the Szemeredi-Trotter curve theorem (actually, a minor generalization of it) of Szekely, applied to certain polygonal arcs.
math.CO↗
Arithmetic structures in random sets
We extend two well-known results in additive number theory, Sárközy's theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restriction-type Fourier analytic argument of Green and Green-Tao.
math.NT↗