arXiv subjects

Gergely Ambrus

Publications and source records attributed to Gergely Ambrus.

At least 19 recordsLinked to original sources

arXiv2026

Isotropic Decompositions via Inverse Eigenvectors

We develop a residue-theoretic framework for studying inverse eigenvectors of a square matrix, defined by the nonlinear equation $M\alpha=\alpha^{-1}$. Our main result is an inverse analogue of the spectral theorem: under natural transversality and properness assumptions, the identity operator admits an explicit decomposition into rank-one tensors associated with the inverse eigenvectors. The proof is based on residues of rational differential forms in several complex variables. For real correlation matrices whose off-diagonal entries have modulus strictly less than one, we remove the properness assumption and show that the coefficients in the decomposition are positive and sum to~$1$. Consequently, the inverse eigenvectors support an explicit centered discrete isotropic probability measure and form a weighted tight frame. We extend the construction to arbitrary real Gram matrices, diagonal inverse eigenvectors, and weighted inverse-eigenvector equations. Simple trace and arithmetic--geometric mean arguments yield inverse eigenvectors with controlled Euclidean norm and coordinate product. These estimates lead to short proofs of the strong real polarization inequality and the $n$th real linear polarization inequality, together with weighted and matrix-valued generalizations and further geometric and analytic applications.

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arXiv2026

Softening locally polyhedral tilings

We call a cell $C \subset \mathbb{R}^d$ soft if every point of its boundary lies on a smooth curve contained in $\partial C$. A tiling of the space is called completely soft if all of its cells are soft. In their 2024 article, Domokos, Goriely, G. Horv\'{a}th and Reg\H{o}s conjectured that every polyhedral tiling of $\mathbb{R}^3$ satisfying mild regularity assumptions can be locally deformed into a completely soft tiling. By constructing an algorithm that first bends the edges emanating from each vertex and then extends this transformation to a sufficiently smooth deformation, they proved the conjecture for polyhedral tilings satisfying a certain combinatorial condition. In the present paper, we precisely describe a new edge-bending algorithm that establishes a more general version of this conjecture: every locally polyhedral tiling of $\mathbb{R}^3$ can be completely softened. We also give a short proof of an earlier result of Domokos, G. Horv\'{a}th, and Reg\H{o}s stating that every suitably nondegenerate polygonic tiling of the plane has, on average, at least two points per cell at which the softness criterion is violated.

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arXiv2025

A note on the Steinitz Lemma

We establish the connection between the Steinitz problem for ordering vector families in arbitrary norms and its variant for not necessarily zero-sum families consisting of `nearly unit' vectors.

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arXiv2025

Covering spiky annuli by planks

Answering Tarski's plank problem, Bang showed in 1951 that it is impossible to cover a convex body $K \subset \mathbb{R}^d$ with $d \geq 1$ by planks whose total width is less than the minimal width $w(K)$ of $K$. In 2003, A. Bezdek asked whether the same statement holds if one is required to cover only the annulus obtained from $K$ by removing a homothetic copy contained within. He showed that if $K$ is the unit square, then saving width in a plank covering is not possible, provided that the homothety factor is sufficiently small. White and Wisewell in 2006 characterized polygons that possess this property. We generalize the constructive part of their classification to spiky convex bodies: a body $K$ is spiky at a boundary point $x$ with supporting hyperplane $H$ and corresponding outer normal $u$, if both $K$ and its tangent cone at $x$ intersect $H$ only at $x$. We show that if $K$ is a convex disc or a convex body in 3-space that is spiky in a minimal width direction, then for every $\varepsilon \in (0,1)$ it is possible to cut a homothetic copy $\varepsilon K$ from the interior of $K$ so that the remaining annulus can be covered by planks whose total width is strictly less than $w(K)$.

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arXiv2025

Large Signed Sums and the Polarization Constant of Convex Bodies

As a counterpart to classical vector balancing, we study large signed sums of unit vectors in finite-dimensional Minkowski spaces and develop their connection with polarization problems for convex bodies. For every convex body $K\subset\mathbb{R}^d$ containing the origin in its interior and every number $n\geq1$ of vectors, we show that the corresponding large signed sum and polarization constants coincide in a common quantity $\mu(K,n)$, which depends only on the symmetric core $K\cap(-K)$. Our main result determines the sharp universal lower bound \[ \mu(K,n) \geq \mu(B_\infty^d,n) = \frac{1}{n}\left\lceil\frac{n}{d}\right\rceil, \] identifying parallelotopes as global minimizers for every $d$ and $n$. In the opposite direction, we prove general upper bounds that show, in particular, that the Euclidean ball is a maximizer up to an absolute constant factor. We also obtain sharp and nearly sharp results in several special cases, including the planar setting. As $n\to\infty$, we prove that $\mu(K,n)$ converges to the Macphail constant of $K$, equivalently to its $1$-absolutely summing constant. Thus, the finite signed-sum problem provides a discrete counterpart of classical Banach space invariants. Combining this connection with sharp results on projection constants, we characterize equality in the corresponding upper bound for the Macphail constant in terms of maximal real equiangular tight frames. Finally, our methods extend to arbitrary vector families, support functions of compact convex sets, and circumradii and diameters of Minkowski sums.

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arXiv2024

Estimates on the decay of the Laplace-Polya integral

The Laplace--P\'olya integral, defined by $J_n(r) = \frac1\pi\int_{-\infty}^\infty \mathrm{sinc}^n t \cos(rt) \mathrm{d} \, t$, appears in several areas of mathematics. We study this quantity by combinatorial methods; accordingly, our investigation focuses on the values at integer $r$'s. Our main result establishes a lower bound for the ratio $\frac{J_n(r+2)}{J_n(r)}$ which extends and generalises the previous estimates of Lesieur and Nicolas, and provides a natural counterpart to the upper estimate established in our previous work. We derive the statement by purely combinatorial, elementary arguments. As a corollary, we deduce that no subdiagonal central sections of the unit cube are extremal, apart from the minimal, maximal, and the main diagonal sections. We also prove several consequences for Eulerian numbers.

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arXiv2024

Non-diagonal critical central sections of the cube

We study the $(n-1)$-dimensional volume of central hyperplane sections of the $n$-dimensional cube $Q_n$. Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular to the main diagonal of the cube is strictly locally maximal for every $n \geq 4$, which was shown before by L. Pournin. Then, we prove that non-diagonal critical central sections of $Q_n$ exist in all dimensions at least $4$. The crux of both proofs is an estimate on the rate of decay of the Laplace-Pólya integral $J_n(r) = \int_{-\infty}^\infty \mathrm{sinc}^n t \cdot \cos (rt) \mathrm{d} t$ that is achieved by combinatorial means. This also yields improved bounds for Eulerian numbers of the first kind.

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arXiv2023

On Helly numbers of exponential lattices

Given a set $S \subseteq \mathbb{R}^2$, define the \emph{Helly number of $S$}, denoted by $H(S)$, as the smallest positive integer $N$, if it exists, for which the following statement is true: for any finite family $\mathcal{F}$ of convex sets in~$\mathbb{R}^2$ such that the intersection of any $N$ or fewer members of~$\mathcal{F}$ contains at least one point of $S$, there is a point of $S$ common to all members of $\mathcal{F}$. We prove that the Helly numbers of \emph{exponential lattices} $\{α^n \colon n \in \mathbb{N}_0\}^2$ are finite for every $α>1$ and we determine their exact values in some instances. In particular, we obtain $H(\{2^n \colon n \in \mathbb{N}_0\}^2)=5$, solving a problem posed by Dillon (2021). For real numbers $α, β> 1$, we also fully characterize exponential lattices $L(α,β) = \{α^n \colon n \in \mathbb{N}_0\} \times \{β^n \colon n \in \mathbb{N}_0\}$ with finite Helly numbers by showing that $H(L(α,β))$ is finite if and only if $\log_α(β)$ is rational.

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arXiv2023

The density of planar sets avoiding unit distances

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set avoiding unit distances cannot exceed $1/4$. Our argument implies the upper bound of $0.2470$.

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arXiv2023

Critical central sections of the cube

We study the volume of central hyperplane sections of the cube. Using Fourier analytic and variational methods, we retrieve a geometric condition characterizing critical sections which, by entirely different methods, was recently proven by Ivanov and Tsiutsiurupa. Using this characterization result, we prove that critical central hyperplane sections in the 3-dimensional case are all diagonal to a (possibly lower dimensional) face of the cube, while in the 4-dimensional case, they are either diagonal to a face, or, up to permuting the coordinates and sign changes, perpendicular to the vector $(1,1,2,2)$. This shows the existence of non-diagonal critical central sections.

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arXiv2023

Piercing the chessboard

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that $h_n = \lceil \frac n 2 \rceil$ for each $n \geq 1$. Studying the piercing problem, we show that $0.7n \leq p_n \leq n-1$ for $n\geq 3$, where the upper bound is conjectured to be sharp. The lower bound is proven by using the linear programming method, whose limitations are also demonstrated.

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arXiv2023

Colorful Vector Balancing

We extend classical estimates for the vector balancing constant of $\mathbb{R}^d$ equipped with the Euclidean and the maximum norms proved in the 1980's by showing that for $p =2$ and $p=\infty$, given vector families $V_1, \ldots, V_n \subset B_p^d$ with $0 \in \sum_{i=1}^n \mathrm{conv}\, V_i$, one may select vectors $v_i \in V_i$ with $ \| v_1 + \ldots + v_n \|_2 \leq \sqrt{d}$ for $p=2$, and $ \| v_1 + \ldots + v_n \|_\infty \leq O(\sqrt{d}) $ for $p = \infty$. These bounds are sharp and asymptotically sharp, respectively, for $n \geq d$. The proofs combine linear algebraic and probabilistic methods with a Gaussian random walk argument.

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arXiv2022

Quantitative Helly-type theorems via sparse approximation

We prove the following sparse approximation result for polytopes. Assume that $Q$ is a polytope in John's position. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $Q \subseteq - 2d^2 \, Q'$. As a consequence, we retrieve the best bound for the quantitative Helly-type result for the volume, achieved by Brazitikos, and improve on the strongest bound for the quantitative Helly-type theorem for the diameter, shown by Ivanov and Naszódi: We prove that given a finite family $\mathcal{F}$ of convex bodies in $\mathbb{R}^d$ with intersection $K$, we may select at most $2 d$ members of $\mathcal{F}$ such that their intersection has volume at most $(c d)^{3d /2} \,\mathrm{vol}\, K$, and it has diameter at most $2 d^2 \,\mathrm{diam} \,K$, for some absolute constant $c>0$.

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arXiv2022

Large signed subset sums

We study the following question: for given $d\geq 2$, $n\geq d$ and $k \leq n$, what is the largest value $c(d,n,k)$ such that from any set of $n$ unit vectors in $\mathbb{R}^d$, we may select $k$ vectors with corresponding signs $\pm 1$ so that their signed sum has norm at least $c(d,n,k)$? The problem is dual to classical vector sum minimization and balancing questions, which have been studied for over a century. We give asymptotically sharp estimates for $c(d,n,k)$ in the general case. In several special cases, we provide stronger estimates: the quantity $c(d,n,n)$ corresponds to the $\ell_p$-polarization problem, while determining $c(d, n, 2)$ is equivalent to estimating the coherence of a vector system, which is a special case of $p$-frame energies. Two new proofs are presented for the classical Welch bound when $n = d+1$. For large values of $n$, volumetric estimates are applied for obtaining fine estimates on $c(d,n,2)$. Studying the planar case, sharp bounds on $c(2, n, k)$ are given. Finally, we determine the exact value of $c(d,d+1,d+1)$ under some extra assumptions.

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arXiv2022

A generalization of Bang's lemma

We prove a common extension of Bang's and Kadets' lemmas for contact pairs, in the spirit of the Colourful Carathéodory Theorem. We also formulate a generalized version of the affine plank problem and prove it under special assumptions. In particular, we obtain a generalization of Kadets' theorem. Finally, we give applications to problems regarding translative and homothetic coverings.

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arXiv2022

The symmetric plank problem, revisited

We present a streamlined proof of K. Ball's symmetric plank theorem in $\mathbb{R}^d$, which solves the affine plank problem raised by Th. Bang for symmetric convex bodies.

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arXiv2021

Uniform tight frames as optimal signals

Non-orthogonal communication is a promising technique for future wireless networks (e.g., 6G and Wi-Fi 7). In the vector channel model, designing efficient non-orthogonal communication schemes amounts to the following extremum problem: \[ \max \min_k \frac{|v_k|^2}{σ^2 + \sum_{l \neq k} \langle v_k, v_l \rangle^2} \] where the maximum is taken among vector systems $(v_k)_1^N \subset \mathbb{R}^d$ satisfying $c_1 \leq |v_k|^2 \leq c_2$ for every $k$, and the parameter $σ>0$ corresponds to the noise of the channel. We show that in the case $σ= 0$, uniform tight frames are the only optimal configurations. We also give quantitative bounds on the optimal capacity of vector channels with relatively small noise.

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arXiv2021

New estimates for convex layer numbers

Starting with a finite point set $X \subset \mathbf{R}^d$, the peeling process repeatedly removes the set of the vertices of the convex hull of the current set. The number of peeling steps required to completely remove $X$ is called the layer number of $X$, denoted by $L(X)$. In the article, we study the layer number of evenly distributed families of point sets contained in $B^d$, the $d$-dimensional unit ball. These sets consist of points in $B^d$ whose minimal distance is asymptotically as large as possible. We show that for a set $X$ belonging to an evenly distributed family, $L(X) \geq Ω(|X|^{1/d})$ holds, with the bound being asymptotically sharp. On the other hand, building on earlier results, we prove that $L(X)\leq O(|X|^{2/d})$ holds for $d\geq 2$, which improves greatly on the current upper bound of $O(|X|^{(d+1)/2d})$ for $d \geq 3$. Finally, we provide a recursive construction of evenly distributed families whose sets satisfy $L(X) = Θ(|X|^{2/d - 1/(d 2^{d-1})})$, showing that our upper bound is nearly tight.

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