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Omer Friedland

Publications and source records attributed to Omer Friedland.

At least 19 recordsLinked to original sources
arXiv2026

Product-profile anti-concentration for block-structured multi-affine polynomials

We establish product-profile anti-concentration, density, and Remez estimates for multi-affine polynomials on the cube. Let $X$ be uniformly distributed on $[0,1]^n$, and let $P:[0,1]^n\to\R$ be a nonconstant multi-affine polynomial of exact degree $d$, with all variables active. For a partition $\cB$ of the variables such that no monomial contains two variables from the same block, define $$ w_d(\cB) = \sum_{\cI\subseteq\cB, \abs{\cI} = d-1} \left(\prod_{B\in\cI}\sqrt{\abs B}\right) \sqrt{\sum_{C\in\cB\setminus\cI}\abs C}. $$ We prove the all-center small-ball estimate $$ \sup_{u\in\R} \Prob\{\abs{P(X)-u}\leρ\} \le Φ_d\left( C_dw_d(\cB)\fracρ{\osc(P)} \right), $$ where $\osc(P)$ is the length of the range of $P$ and $Φ_d(t) = t\sum_{j = 0}^{d-1}\log^j(1/t)/j!$ on $(0,1]$, capped at one. Every multi-affine polynomial admits the singleton partition, which yields the universal scale $n^{d-1/2}$; an admissible partition into at most $q$ blocks yields the improved scale $q^{(d-1)/2}n^{d/2}$. If $\cB$ has exactly $d$ blocks, then $$ w_d(\cB) = d\prod_{B\in\cB}\sqrt{\abs B}, $$ and products of centered block averages show that both the small-ball profile and the dependence on the full block-size vector are optimal, up to constants depending only on $d$. We also prove that $P(X)$ has a density $f_P$ satisfying $$ \norm{f_P}_{L^p(\R)} \le C_dp^{d-1} \left( \frac{w_d(\cB)}{\osc(P)} \right)^{1-1/p}, \quad 1<p<\infty, $$ with matching $p$- and block-scale growth for $p\ge2$ on the block-product models. As an application, we derive translation-invariant quotient Remez inequalities with the same structural scale. The proof combines weighted block selection, affine cube slicing, degree-lowering contractions, and the exact recursion underlying the product profile.

math.CA↗
arXiv2026

A Log-Free Lower Bound for the Number of Facets of $0/1$-Polytopes

Let $g(n)$ denote the largest number of facets of a full-dimensional $0/1$-polytope in $\R^n$. We prove that there are absolute constants $c>0$ and $n_0$ such that $$ g(n)\ge (cn)^{n/2}\quad(n\ge n_0). $$ This removes the logarithmic factor from the lower bound $\bigl(cn/\log n\bigr)^{n/2}$ of Gatzouras, Giannopoulos, and Markoulakis. The proof compares a random sign polytope with two Rademacher rate bodies separated by a fixed level gap. Facets missing the inner body have uniformly small footprints on a flat patch of the outer body. A facet entering the inner body forces an empty buffered discrete cap. For shallow penetration, a likelihood-slab localization reduces the relevant range entropy and permits a conditional $\varepsilon$-net argument; for deep penetration, a global discretization suffices.

math.CO↗
arXiv2026

A $5/8$ Lower Bound on the Banach-Mazur Distance to the Cross-Polytope

Let $Γ$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m = Γ(B_1^m)$ be the associated Gaussian Gluskin polytope. In the regime $m = n^3$ we prove that, with probability at least $1-C/n$, $$ d_{\mathrm{BM}}(G_m,B_1^n) \ge c n^{5/8}(\log n)^{-1/4}. $$ This improves the polynomial exponent $4/7$ obtained in the author's preceding work and gives an explicit logarithmic factor. The proof retains the discretization and conditioning/powering framework, but replaces the earlier split into two coefficient regimes by two uniform quotient events. One controls successive directions of the big-coordinate parts; the other compresses the entire small-coordinate cloud near a low-dimensional subspace after every admissible quotient. Suppression, a local Maurey argument, and Gram-Schmidt volume estimates then combine these two forms of control.

math.PR↗
arXiv2026

A Disk-Growth Remez Principle and a Modular Proof of the Measurable Turán-Nazarov Inequality

We give a modular proof of the measurable Turán-Nazarov inequality for exponential polynomials. The proof first establishes a Remez principle for holomorphic functions satisfying two disk-growth assumptions. The global growth assumption controls the number of relevant zeros, while the local growth assumption gives an effective degree. This yields Cartan coverings, sublevel estimates, and a geometric-mean Remez inequality. For exponential polynomials with bounded spectral diameter, the required disk growth follows from the classical interval Turán inequality. For large spectral diameter, we use a first-order pruning step. If $ρ= \diam(\spec p)$ and $a\in\spec p$, then $$ Q_a = ρ^{-1}(D-a)p $$ has one fewer exponential term, and the quotient $Q_a/p$ satisfies an absolute weak distribution estimate away from the zero set of $p$. Writing $$ Q_a = ρ^{-1}(D-a)p, \quad Q_b = ρ^{-1}(D-b)p $$ for two farthest spectral points $a,b$ gives $$ Q_a-Q_b = \frac{b-a}ρp, \quad |b-a| = ρ, $$ and hence $|p|\le |Q_a|+|Q_b|$. The induction is carried out in geometric-mean form on the original measurable set. This avoids losing a fixed proportion of the set at each step and gives the classical measurable Turán-Nazarov inequality with the sharp algebraic exponent $m-1$. The final measurable $L^\infty$ estimate is classical; the point here is the modular proof and the geometric-mean induction. The only Turán-type input is the classical interval Turán inequality.

math.CV↗
arXiv2026

On the inner radius of the nonvanishing set for eigenfunctions of complex elliptic operators

Let $Ω\subset\mathbb{R}^d$ be any open set. We consider solutions of $Hψ_λ=λψ_λ$, $λ\in\mathbb{C}$, where $H$ is an $m$th order complex constant-coefficient elliptic partial differential operator. We prove that either the eigenfunctions satisfy a lower bound on the inner radius of the complement of the zero set of $ψ_λ$ in $Ω$ of order $|λ|^{-1/m}$, or 100% of the $L^2$ mass of $ψ_λ$ concentrates in a boundary layer of width $|λ|^{-1/m}$, as $|λ|\to+\infty$.

math.AP↗
arXiv2026

An improved lower bound on the Banach--Mazur distance to the cross-polytope

Let $Γ$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m = Γ(B_1^m)$ be the associated Gaussian Gluskin polytope (equivalently, a random $n$-dimensional quotient of $\ell_1^m$). In the regime $m = n^3$ we prove that, with probability at least $1-2/n$, $$ d_{\mathrm{BM}}(G_m,B_1^n) \ge c n^{4/7}(\log n)^{-C}, $$ where $B_1^n = \conv\{\pm e_1,\dots,\pm e_n\}$ is the cross-polytope. This improves the previously best-known exponent $5/9$ (up to logarithmic factors) for this Gaussian model; in particular, the same lower bound holds for $\sup_{K} d_{\mathrm{BM}}(K,B_1^n)$. The main new ingredient is a conditioning-compatible treatment of the regime of ``many small-coefficients''. After passing to a suitable Gaussian quotient, we apply a Maurey-type sparsification that reduces the relevant entropy (in effect shrinking the support size from $k$ to $k/\log(nρ)$) at the cost of a Euclidean thickening. We control this enlargement via a Gaussian measure bound stable under Euclidean thickening. In the complementary regime of ``few small-coefficients'', we give a streamlined argument avoiding the global tilting step in earlier work. Together these ingredients rebalance entropy and small-ball estimates and yield the exponent $4/7$.

math.FA↗
arXiv2026

Geometry-Driven Conditioning of Multivariate Vandermonde Matrices in High-Degree Regimes

We study multivariate monomial Vandermonde matrices $V_N(Z)$ with arbitrary distinct nodes $Z=\{z_1,\dots,z_s\}\subset B_2^n$ in the high-degree regime $N\ge s-1$. Introducing a projection-based geometric statistic -- the \emph{max-min projection separation} $ρ(Z,j)$ and its minimum $κ(Z)=\min_jρ(Z,j)$ -- we construct Lagrange polynomials $Q_j\in\mathcal P_N^n$ with explicit coefficient bounds $$ \|Q_j\|_\infty \lesssim s\Bigl(\frac{4n}{ρ(Z,j)}\Bigr)^{s-1}. $$ These polynomials yield quantitative distance-to-span estimates for the rows of $V_N(Z)$ and, as consequences, $$ σ_{\min}(V_N(Z)) \gtrsim \frac{κ(Z)^{s-1}}{(4n)^{s-1} s\sqrt{s ν(n,N)}}, \quad ν(n,N)={N+n\choose N}, $$ and an explicit right inverse $V_N(Z)^+$ with operator-norm control $$ \|V_N(Z)^+\| \lesssim s^{3/2}\sqrt{ν(n,N)}\Bigl(\frac{4n}{κ(Z)}\Bigr)^{s-1}. $$ Our estimates are dimension-explicit and expressed directly in terms of the local geometry parameter $κ(Z)$; they apply to \emph{every} distinct node set $Z\subset B_2^n$ without any \emph{a priori} separation assumptions. In particular, $V_N(Z)$ has full row rank whenever $N\ge s-1$. The results complement the Fourier-type theory (on the complex unit circle/torus), where lower bounds for $σ_{\min}$ hinge on uniform separation or cluster structure; here stability is quantified instead via high polynomial degree and the projection geometry of $Z$.

math.CA↗
arXiv2025

Local $l^\infty$ bounds for eigenfunctions of complex elliptic operators via diophantine problems

We prove local bounds on the amplitude of eigen- functions of complex constant-coefficient elliptic operators with a smooth potential on an arbitrary open subset of \R^d by estimating it in terms of the number of solutions of a diophantine inequality arising from the symbol of the operator. In the special case of positive elliptic operators, we recover H örmander's classical exponent up to an arbitrarily small loss. We show that a much better exponent may be obtained when the principal symbol of the oper- ator has complex coefficients. We generalize our estimate to any higher-order derivatives of eigenfunctions.

math.AP↗
arXiv2019

Superscarred quasimodes on flat surfaces with conical singularities

We construct a continuous family of quasimodes for the Laplace-Beltrami operator on a translation surface. We apply our result to rational polygonal quantum billiards and thus construct a continuous family of quasimodes for the Neumann Laplacian on such domains with spectral width ${\mathcal{O}}_\varepsilon(λ^{3/8+\varepsilon})$. We show that the semiclassical measures associated with this family of quasimodes project to a finite sum of Dirac measures on momentum space, hence, they satisfy Bogomolny and Schmit's superscar conjecture for rational polygons.

math.FA↗
arXiv2019

Exponential Taylor domination

Let $f(z) = \sum_{k=0}^\infty a_k z^k$ be an analytic function in a disk $D_R$ of radius $R>0$, and assume that $f$ is $p$-valent in $D_R$, i.e. it takes each value $c\in{\mathbb C}$ at most $p$ times in $D_R$. We consider its Borel transform $$ B(f)(z) = \sum_{k=0}^\infty \frac{a_k}{k!} z^k , $$ which is an entire function, and show that, for any $R>1$, the valency of the Borel transform $B(f)$ in $D_R$ is bounded in terms of $p,R$. We give examples, showing that our bounds, provide a reasonable envelope for the expected behavior of the valency of $B(f)$. These examples also suggest some natural questions, whose expected answer will strongly sharper our estimates. We present a short overview of some basic results on multi-valent functions, in connection with "Taylor domination", which, for $f(z) = \sum_{k=0}^\infty a_k z^k$, is a bound of all its Taylor coefficients $a_k$ through the first few of them. Taylor domination is our main technical tool, so we also discuss shortly some recent results in this direction.

math.CA↗
arXiv2019

Doubling coverings via resolution of singularities and preparation

In this paper we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form $$ κ({\mathcal U}) \le K_1(\log ({1}/δ))^{K_2} . $$ This is done in a rather general setting, i.e. for the $δ$-complement of a polynomial zero-level hypersurface $Y_0$ and for the regular level hypersurfaces $Y_c$ themselves with no assumptions on the singularities of $P$. The coefficient $K_2$ is the ambient dimension $n$ in the first case and $n-1$ in the second case. However, the question of a uniform behavior of the coefficient $K_1$ remains open. As a second theme, we confirm in arbitrary dimension the upper bound for the number of a-charts covering a real semi-algebraic set $X$ of dimension $m$ away from the $δ$-neighborhood of a lower dimensional set $S$, with bound of the form $$ κ(δ) \le C (\log ({1}/δ))^{m} $$ holding uniformly in the complexity of $X$. We also show an analogue for level sets with parameter away from the $δ$-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

math.CA↗
arXiv2018

Scarred quasimodes on translation surfaces

Rational polygonal billiards are one of the key models among the larger class of pseudo-integrable billiards. Their billiard flow may be lifted to the geodesic flow on a translation surface. Whereas such classical billiards have been much studied in the literature, the analogous quantum billiards have received much less attention. This paper is concerned with a conjecture of Bogomolny and Schmit who proposed in 2004 that the eigenfunctions of the Laplacian on rational polygonal billiards ought to become localized along a finite number of vectors in momentum space, as the eigenvalue tends to infinity. For any given momentum vector $ξ_0\in\mathbb{S}^1$ we construct a continuous family of quasimodes which gives rise to a semi-classical measure whose projection on momentum space is supported on the orbit $Dξ_0$, where $D$ denotes the dihedral group associated with the rational polygon.

math-ph↗
arXiv2017

Doubling chains on complements of algebraic hypersurfaces

A doubling chart on an $n$-dimensional complex manifold $Y$ is a univalent analytic mapping $ψ:B_1\to Y$ of the unit ball in $\mathbb{C}^n$, which is extendible to the (say) four times larger concentric ball of $B_1$. A doubling covering of a compact set $G$ in $Y$ is its covering with images of doubling charts on $Y$. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney's ball coverings of a domain $W\subset {\mathbb R}^n$, introduced in [17] (compare [9]). We study doubling chains in the complement $Y=\mathbb{C}^n\setminus H$ of a complex algebraic hypersurface $H$ of degree $d$ in $\mathbb{C}^n$, and provide information on their length and other properties. Our main result is that any two points $v_1,v_2$ in a distance $δ$ from $H$ can be joined via a doubling chain in the complement $Y=\mathbb{C}^n\setminus H$ of length at most $c_1\log (\frac{c_2}δ)$ with explicit constants $c_1,c_2$ depending only on $n$ and $d$. As a consequence, we obtain an upper bound on the Kobayashi distance in $Y$, and an upper bound for the constant in a doubling inequality for regular algebraic functions on $Y$. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on $Y$.

math.CA↗
arXiv2017

Approximating matrices and convex bodies through Kadison-Singer

We show that any $n\times m$ matrix $A$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $A$. This improves on existing results by removing an extra logarithmic factor in the size of the extracted matrix. Our proof uses the recent solution of the Kadison-Singer problem. We also develop a sort of tensorization technique to deal with constraint approximation problems. As an application, we provide a sparsification result with equal weights and an optimal approximate John's decomposition for non-symmetric convex bodies. This enables us to show that any convex body in $\mathbb{R}^n$ is arbitrary close to another one having $O(n)$ contact points and fills the gap left in the literature after the results of Rudelson and Srivastava by completely answering the problem. As a consequence, we also show that the method developed by Guédon, Gordon and Meyer to establish the isomorphic Dvoretzky theorem yields to the best known result once we inject our improvements.

math.FA↗
arXiv2016

Doubling coverings of algebraic hypersurfaces

A doubling covering $\U$ of a complex $n$-dimensional manifold $Y$ consists of analytic functions $ψ_j:B_1\to Y$, each function being analytically extendable, as a mapping to $Y$, to a four times larger concentric ball $B_4$. Main result of this paper is an upper bound on the minimal number $κ({\U})$ of charts in doubling coverings of a manifold $Y$, being a compact part of a non-singular level hypersurface $Y=\{P=c\}$, where $P$ is a polynomial on $\C^n$ with non-degenerated critical points. We show that $κ({\U})$ is of order $\log({1}/ρ)$, where $ρ$ is the distance from $Y$ to the singular set of $P$. Our main motivation is that doubling coverings form a special class of "smooth parameterizations", which are used in bounding entropy type invariants in smooth dynamics on one side, and in bounding density of rational points in diophantine geometry on the other. Complexity of smooth parameterizations is a key issue in some important open problems in both areas. We also present connections between doubling coverings and doubling inequalities for analytic functions $f$ on $Y$, which compare the maxima of $|f|$ on couples of compact domains $Ω\subset G$ in $Y$. We shortly indicate connections with Kobayashi metric and with Harnack inequality.

math.CA↗
arXiv2015

Inverse Littlewood-Offord problems for Quasi-Norms

Given a star-shaped domain $K\subseteq \mathbb R^d$, $n$ vectors $v_1,\dots,v_n \in \mathbb R^d$, a number $R>0$, and i.i.d. random variables $η_1,\dots,η_n$, we study the geometric and arithmetic structure of the set of vectors $V = \{v_1,\dots,v_n\}$ under the assumption that the small ball probability \[\sup_{x\in \mathbb R^d}~\mathbb P\Bigg(\sum_{j=1}^nη_jv_j\in x+RK\Bigg)\] does not decay too fast as $n\to \infty$. This generalises the case where $K$ is the Euclidean ball, which was previously studied by Nguyen-Vu and Tao-Vu.

math.PR↗
arXiv2015

Small ball estimates for quasi-norms

This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood-Offord type estimates for quasi-norms. This generalizes a result which was previously obtained by Friedland and Sodin and by Rudelson and Vershynin.

math.PR↗