arXiv subjects
Publications and source records attributed to Swaroop Hegde.
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 $r$.
Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--S\'ark\"ozy 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.
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 $|A+A+A| \geq \frac{1}{M} |A+A|^{3/2}$ 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 $1.$