Analysis Seminar, Dept. of Mathematics and Applied Math. -- Σεμινάριο Ανάλυσης

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Analysis Seminar in Crete (2026-27)

Abstract: For $N$ lattices in $\RR^d$ with volume $1$ and pairwise trivial intersections, every nonzero common tiling function has support diameter $\Omega(N^{1/d})$, while for lattice families whose fundamental domains have uniformly bounded diameters, the standard convolution construction gives an $O(N)$ upper bound, leaving a gap that has remained open since the work of Kolountzakis and Wolff. We close this gap by constructing, for every $d\geq 2$ and all $N\ge 2$, lattice families satisfying the same volume and intersection conditions that admit a nonnegative common tiling function with support diameter $O(N^{1/d})$, thereby also answering a question posed by Kolountzakis and Papageorgiou. We also obtain the optimal $O(\sqrt N)$ upper bound by constructing, for any prescribed family of plane lattices whose volumes lie in a fixed bounded set independent of $N$, a pairwise trivially intersecting family with the same respective volumes and with bases arbitrarily close to suitable bases of the prescribed lattices.

Abstract: We deal with the existence of bounded and measurable common fundamental domains for two lattices in $\mathbb{R}^d$ as well as three fully commensurable lattices, i.e. lattices whose sum is again a lattice. The concept of a fundamental domain of a lattice $L\subseteq \mathbb{R}^d$ is similar to a set that tiles $\mathbb{R}^d$ by translations of $L$. In fact every fundamental domain of a lattice, tiles $\mathbb{R}^d$. While the converse is not generally true, sets that tile the space by lattice translations behave as fundamental domains for most practical applications. We show that for any two lattices of equal volume, there always exists a bounded and measurable common fundamental domain. This generalizes the result by S. Grepstad and M. Kolountzakis which states that for any two lattices of equal volume there exists a bounded and measurable set that tiles $\RR^d$ with both lattices. As for two lattices of unequal volume, $L, M\subseteq \RR^d$, answering a question raised by Han and Wang, we show that if ${\rm vol} (L)<{\rm vol} (M)$ then there exists a bounded and measurable set that tiles $\RR^d$ by translations of $L$ and packs $\RR^d$ by translations of $M$. Finally, we completely characterize when three subgroups of equal order in a finite abelian group share a common transversal. As a consequence, we determine precisely when three full-rank lattices of equal volume with a discrete sum in $\RR^d$ admit a common fundamental domain, thereby answering an open question from 1997 for the case $n=3$.

Thu, 08 Oct 2026, 11:15, Room: A303
Speaker: Stefanos Koustas (University of Athens)

Abstract: The Mizohata-Takeuchi conjecture was recently disproved by Cairo and Zhang (2025), shifting the focus away from its classical and well-known setting. In this talk, we establish a new local Mizohata-Takeuchi estimate for Hörmander-type operators with non-degenerate phases, showing an explicit dependence on the phase's signature. Our result is driven by refined decoupling estimates—originating from the work of Guth, Iosevich, Ou, Wang adapted to this broader class of operators and generalizes previous work of Carbery, Iliopoulou, Wang (2023), which treats the maximal signature case.