arXiv subjects

Nagendar Reddy Ponagandla

Publications and source records attributed to Nagendar Reddy Ponagandla.

4 recordsLinked to original sources

arXiv2026

Quasipolynomial density bounds for $K$-point configurations in $\mathbb{Z}^d$

Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $Δ\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $Δ$. We prove that \[ |A|\ll_{Δ,d} N^d\exp\!\left(-c_{Δ,d}\sqrt{\log N}\right) \] improving upon a polylogarithmic bound due to Magyar. We perform a density increment argument using the circle method, and we introduce a ``cut operator'' method to decouple the weighted exponential sum over the system of quadratic forms describing the simplex. Our proof combines ideas from graph theory, functional analysis, and the geometry of numbers. In the process, we apply Finner's fractional form of Hölder's inequality, the analytic large sieve, and Kim's mean value formula for primitive lattice flags.

math.NT↗

arXiv2026

Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality

Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials $h\in\mathbb{Z}[x]$. In particular, let \[ D(h(\mathbb{N}),X):= \max{|A|:\ A\subseteq [1,X]\cap\mathbb{N} \text{and}\ (A-A)\cap h(\mathbb{N})\subseteq\{0\}}. \] We prove that for every $0<μ<1/2$ there are constants $c_0, X_{\text{min}}>0$ depending on $h$ and $μ$ such that for every $X>X_{\text{min}}$, \[D(h(\mathbb{N}), X)\leq Xe^{-c_0(\log X)^μ}.\] This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range $0<μ<1/2$, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the ``random sparsification'' procedure of Green and Sawhney.

math.NT↗

arXiv2026

Refined upper bounds on Schur-like numbers

For positive integers $r, m$ and $N$, every $r$-coloring of $\{1, \dots, N\}$ contains a monochromatic solution to $x_1+\dots+x_{m+1}=y_1+\dots+y_m$ provided that $N \ge 3^r (r!)^{1/m}$, which is qualitatively optimal when $m$ is logarithmic in $r$.

math.CO↗

arXiv2026

Polynomial progressions in the generalized twin primes

By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\leq 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. Let $P_1,...,P_t\in \mathbb{Z}[y]$ with $P_1(0)=\cdots=P_t(0)=0$. We use the transference argument of Tao and Ziegler to prove there exist positive integers $x, y,$ and $b \leq 246 $ such that $x+P_1(y),x+P_2(y),...,x+P_t(y)$ and $x+P_1(y)+b,x+P_2(y)+b,...,x+P_t(y)+b$ are all prime. Our work is inspired by Pintz, who proved a similar result for the special case of arithmetic progressions.

math.NT↗