We give a detailed account of the recent work of Hickman and Srivastava on counting integral points in a $ \delta $-neighbourhood of a dilate $ q \mathcal{M} $ of a non-degenerate curve $ \mathcal{M} \subset \mathbb{R}^{n} $, $ n \ge 3 $. Our aim is expository: we simplify the proof, provide additional details, refine the statement of the theorem and provide calculations from a more fundamental perspective. We also formulate a new conjecture based on recent work of Chen, Seeger, Srivastava and Technau.
We prove that every compact connected Riemannian isometric filling $M$ of a circle of length $2\pi$ satisfies $\operatorname{Area}(M) \geq \frac{14\zeta(3)}{\pi} \approx 5.35677$, regardless of orientability or topological types. Our new approach uses the odd Fourier coefficients of the distance functions from boundary...
We describe, with proof, an infinite class of examples of finite sets of integers $A$ for which $\|\widehat{1_A}\|_1\geq (c_\mathcal{R}-o_{|A| \rightarrow \infty}(1))\sqrt{|A|}$ where $c_\mathcal{R}=0.953\dots$.
In this paper, after reviewing recent progress on the cohomology of $\overline{{\cal M}}_{0,n}$, we further our investigation on the cohomology of moduli spaces of pointed curves in continuation of [2,4,5,6,7,8]. In particular, we prove that the Betti number distribution of the Fulton-MacPherson compactification $C[n]$...
We develop a theory of logarithmic $A_{\mathrm{inf}}$-cohomology with coefficients for a class of $p$-adic log formal schemes that are ``sufficiently log smooth'', where the coefficients are given by relative log BKF modules. Then we establish comparison isomorphisms with \'etale, de Rham, and crystalline cohomology, a...
Let $f,g:\mathbb{N}\to\mathbb{T}$ be completely multiplicative functions with dense images in the complex unit circle $\mathbb{T}$. We prove that, for every non-empty open set $U \subset \mathbb{T}^2$, the set of integers $n$ such that $(f(n),g(n+1)) \in U$ has positive lower logarithmic density, unless the pair $(f,g)...
Let $X$ be a finite Euclidean Ramsey set. We prove that adjoining any point outside the affine hull of $X$ gives another Euclidean Ramsey set, answering a conjecture of Ivan, Leader, and Walters. We first give an elementary product proof under the additional assumptions that the orthogonal projection of the new point l...
Mostafa Mirabi· 0 citations
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