We prove that $(\alpha n\log n)_{n\ge1}$ has Poissonian pair correlation for almost every real $\alpha$, solving a Problem posed by Aistleitner, El-Baz and Munsch [Open Problem 3, Geom. Funct. Anal. 2021]. Our main contribution is a new four-point counting estimate with a fixed power saving over $N^{7/3}$ at tolerance $N^{-1}$, valid on the ranges left after a first-moment and Borel--Cantelli argument. Its proof reduces the counting condition to the required proximity of an implicitly defined real root to an integer; a Fej\'er kernel majorizes this condition, and multidimensional van der Corput $A$- and $B$-processes supply the saving. A localized Mellin argument then yields the second-moment estimate needed for the metric conclusion.
Let $E(X)=\#\{(a,b)\in\Z^2:a^2+b^2\le X\}-\pi X$. We prove that $E(X)=O_\varepsilon(X^{209/670+\varepsilon})$ for every $\varepsilon>0$, uniformly for real $X\ge4$. This breaks the $5/16$ threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements:...
Xi Cen, Xi-Tao Gao, Cheng Zhang et al.· 0 citations
We prove that any subset $A \subseteq \{1, \dots, N\}$ lacking non-trivial polynomial progressions $(x, x + y, x + y^2)$ satisfies $\lvert A \rvert \le CN \exp\bigl(-c(\log \log N)^\beta\bigr)$ for effective absolute constants $C, c, \beta>0$. The argument develops a density increment framework localized to anisotropic...
Let $1\leq k\leq n$. We prove that $k$-dimensional intrinsic Lipschitz graphs in the Heisenberg group $\mathbb{H}^n$ satisfy a geometric lemma $\mathrm{GLem}(\beta_{2,\mathcal{V}_k},p)$ for horizontal $\beta$-numbers with an exponent $p=p(k)$. Previously, this result was known only in the case $k=1$; our proof recovers...
Yi-Bo Chen, Katrin Fassler, Kilian Zambanini University of Jyväskylä et al.· 0 citations
Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from th...
Let $L_v=-\Delta-2v\cdot\nabla$ on $L^2(\mathbb R^n,e^{2v\cdot x}\,dx)$, where $v\in\mathbb R^n\setminus\{0\}$ is a constant drift vector. We prove that the full vector Riesz transform $\nabla L_v^{-1/2}$ satisfies a weak-type $(1,1)$ estimate for real-valued functions with constant at most $2$, uniformly in both the d...
For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}....
Mayukh Mukherjee· 0 citations
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