Publications

2026

1. arXiv

   A somewhat sure note on an un-Schur problem

   Swaroop Hegde, Hitesh Kumar, and Pratibha

   2026

   Abs arXiv

   Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound is optimal. We disprove this conjecture and prove new bounds. In particular, we show that the maximum fraction of rainbow Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ lies between $9/22$ and $8/15$ asymptotically. Moreover, we study the problem in the general $k$-color setting and establish new non-trivial bounds.

2. arXiv

   Refined upper bounds on Schur-like numbers

   Swaroop Hegde, Andrew Lott, Giorgis Petridis, and 1 more author

   2026

   Abs arXiv

   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 $r$.

3. arXiv

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

   Carlo Francisco E. Adajar, Rishika Agrawal, Mukul Rai Choudhuri, and 6 more authors

   2026

   Abs arXiv

   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 $0<\mu<1/2$ there are constants $c_0, X_{\text{min}}>0$ depending on $h$ and $\mu$ such that for every $X>X_{\text{min}}$, \[D(h(\mathbb{N}), X)\leq Xe^{-c_0(\log X)^\mu}.\] 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 $0<\mu<1/2$, 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.

2025

1. arXiv

   

   An inverse and a stability result for Ruzsa's inequality on triple sumsets

   Swaroop Hegde

   2025

   Abs arXiv

   Ruzsa's inequality states that $|A+A+A| \leq |A+A|^{3/2}$ for any finite set $A$ in a commutative group. Ruzsa has constructed examples showing that this inequality is sharp asymptotically, up to a constant factor. We prove an inverse result which says that if \(|A+A+A| \geq \frac{1}{M} |A+A|^{\frac{3}{2}} \) for some parameter $M,$ then the set $A$ resembles the sets in Ruzsa's construction. We then construct more families of examples which suggest that our inverse result is likely best possible qualitatively. The method extends to give an inverse result for a higher sumset analogue of Ruzsa's inequality, namely $|(h+1)A| \leq |hA|^{\frac{h+1}{h}}$ for any $h\geq 2.$ We also provide a ``99%-stability" version of Ruzsa's inequality, which describes near optimal structures when $M$ is very close to $1.$