arXiv subjects

Souktik Roy

Publications and source records attributed to Souktik Roy.

5 recordsLinked to original sources

arXiv2022

Semialgebraic methods and generalized sum-product phenomena

For a bivariate $P(x,y) \in \mathbb{R}[x,y]\setminus (\mathbb{R}[x] \cup \mathbb{R}[y])$, our first result shows that for all finite $A \subseteq \mathbb{R}$, $|P(A,A)|\geq α|A|^{5/4}$ with $α=α(\mathrm{deg} P) \in \mathbb{R}^{>0}$ unless $$ P(x,y)=f(γu(x)+δu(y)) \text{ or } P(x,y)=f(u^m(x)u^n(y)) $$ for some univariate $f, u \in \mathbb{R}[t]\setminus \mathbb{R}$, constants $γ, δ\in \mathbb{R}^{\neq 0}$, and $m, n\in \mathbb{N}^{\geq 1}$. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results are sum-product type theorems for two polynomials, generalizing the classical result by Erdos and Szemerédi as well as a theorem by Shen. We also obtained similar results for $\mathbb{C}$, and from this deduce results for fields of characteristic $0$ and fields of large prime characteristic. The proofs of our results use tools from semialgebraic/o-minimal geometry.

math.LO↗

arXiv2021

An upper bound on the size of Sidon sets

In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.

math.CO↗

arXiv2019

Expanding Polynomials and Pairs of Polynomials in Characteristic 0

We begin a generalized study of sum-product type phenomenon in different fields by considering pairs $P(x,y)$ and $Q(x,y)$ of two variable polynomials that simultaneously exhibit small symmetric expansion. Our first result is that such $P(x,y)$ and $Q(x,y)$ over $\mathbb{R}$ and $\mathbb{C}$ have very similar structure, obtained by employing semi-algebraic geometry/o-minimality. Then using model-theoretic transfer and basic Galois theory we deduce results for fields of characteristic $0$ and characteristic $p$ when $p$ is large. We obtain as corollaries a generalization of Elekes-Rónyai type structural results to arbitrary characteristic 0 fields, and a strengthening of these classic results in a symmetric case of natural interest. We note a related bound of $5/4$ in the exponent for the sum-product problem in finite fields of large characteristic, although a lower bound for this characteristic cannot be computed from our methods.

math.CO↗

arXiv2019

Small doublings in abelian groups of prime power torsion

Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In this paper, we confirm the conjecture when $r$ is a power of prime. In particular, the bound we obtain is tight.

math.CO↗

arXiv2017

Non-optimality of the Greedy Algorithm for subspace orderings in the method of alternating projections

The method of alternating projections involves projecting an element of a Hilbert space cyclically onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm and that one can obtain estimates for the rate of convergence in terms of quantities describing the geometric relationship between the subspaces in question, namely their pairwise Friedrichs numbers. We consider the question of how best to order a given collection of subspaces so as to obtain the best estimate on the rate of convergence. We prove, by relating the ordering problem to a variant of the famous Travelling Salesman Problem, that correctness of a natural form of the Greedy Algorithm would imply that $\mathrm{P}=\mathrm{NP}$, before presenting a simple example which shows that, contrary to a claim made in the influential paper [Kayalar-Weinert, Math. Control Signals Systems, vol. 1(1), 1988], the result of the Greedy Algorithm is not in general optimal. We go on to establish sharp estimates on the degree to which the result of the Greedy Algorithm can differ from the optimal result. Underlying all of these results is a construction which shows that for any matrix whose entries satisfy certain natural assumptions it is possible to construct a Hilbert space and a collection of closed subspaces such that the pairwise Friedrichs numbers between the subspaces are given precisely by the entries of that matrix.

math.NA↗