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1. Characterizing Schwarz maps by tracial inequalities

   https://doi.org/10.1007/s11005-023-01636-4  

   Carlen, Eric; Müller-Hermes, Alexander (February 2023, Letters in Mathematical Physics)

   Abstract Let$$\phi $$ $\varphi$be a positive map from the$$n\times n$$ $n\times n$matrices$$\mathcal {M}_n$$ $M_n$to the$$m\times m$$ $m\times m$matrices$$\mathcal {M}_m$$ $M_m$. It is known that$$\phi $$ $\varphi$is 2-positive if and only if for all$$K\in \mathcal {M}_n$$ $K\in M_n$and all strictly positive$$X\in \mathcal {M}_n$$ $X\in M_n$,$$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$ $\varphi \left(K^{\ast }{X}^{-1}K\right)⩾\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)$. This inequality is not generally true if$$\phi $$ $\varphi$is merely a Schwarz map. We show that the corresponding tracial inequality$${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$ $\text{Tr}\left[\varphi \left(K^{\ast }{X}^{-1}K\right)\right]⩾\text{Tr}\left[\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)\right]$holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. 

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2. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

   https://doi.org/10.1007/s00224-022-10106-8  

   Vyas, Nikhil; Williams, R_Ryan (November 2022, Theory of Computing Systems)

   Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$ $\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[n^{{\left(\log n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$$\mathcal { C}$$ $C$, by showing that$$\mathcal { C}$$ $C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class$${\mathcal C}$$ $C$. Say that a symmetric Boolean functionf(x₁,…,x_n) issparseif it outputs 1 onO(1) values of$${\sum }_{i} x_{i}$$ ${\sum }_i{x}_i$. We show that for every sparsef, and for all “typical”$$\mathcal { C}$$ $C$, faster#SAT algorithms for$$\mathcal { C}$$ $C$circuits imply lower bounds against the circuit class$$f \circ \mathcal { C}$$ $f\circ C$, which may bestrongerthan$$\mathcal { C}$$ $C$itself. In particular:#SAT algorithms forn^k-size$$\mathcal { C}$$ $C$-circuits running in 2ⁿ/n^ktime (for allk) implyNEXPdoes not have$$(f \circ \mathcal { C})$$ $(f\circ C)$-circuits of polynomial size.#SAT algorithms for$$2^{n^{{\varepsilon }}}$$ $2^{n^{\epsilon }}$-size$$\mathcal { C}$$ $C$-circuits running in$$2^{n-n^{{\varepsilon }}}$$ $2^{n-n^{\epsilon }}$time (for someε> 0) implyQuasi-NPdoes not have$$(f \circ \mathcal { C})$$ $(f\circ C)$-circuits of polynomial size. Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC⁰∘THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC⁰[Williams JACM’14] andACC⁰∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class. 

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3. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven_J; Romero_Acosta, Francisco; et al (March 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$, let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$, where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$. Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$. 

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4. Dimension-free discretizations of the uniform norm by small product sets

   https://doi.org/10.1007/s00222-024-01306-9  

   Becker, Lars; Klein, Ohad; Slote, Joseph; Volberg, Alexander; Zhang, Haonan (December 2024, Inventiones mathematicae)

   Abstract Let$$f$$ $f$be an analytic polynomial of degree at most$$K-1$$ $K-1$. A classical inequality of Bernstein compares the supremum norm of$$f$$ $f$over the unit circle to its supremum norm over the sampling set of the$$K$$ $K$-th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for$$L^{p},1\le p\leq \infty $$ $L^p,1\le p\le \infty$norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of$$\deg (f)$$ $deg\left(f\right)$and is instead governed by the maximumindividualdegree of$$f$$ $f$;i.e., the largest degree of$$f$$ $f$when viewed as a univariate polynomial in any coordinate. For example, we find that for$$n$$ $n$-variate analytic polynomials$$f$$ $f$of degree at most$$d$$ $d$and individual degree at most$$K-1$$ $K-1$,$$\|f\|_{L^{\infty }(\mathbf{D}^{n})}\leq C(X)^{d}\|f\|_{L^{\infty }(X^{n})}$$ ${\parallel f\parallel }_{L^{\infty }\left(D^n\right)}\le C{\left(X\right)}^d{\parallel f\parallel }_{L^{\infty }\left(X^n\right)}$for any fixed$$X$$ $X$in the unit disc$$\mathbf{D}$$ $D$with$$|X|=K$$ $\left|X\right|=K$. The dependence on$$d$$ $d$in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant$$\mathcal{O}(\log K)^{2d}$$ $O{(logK)}^{2d}$. 

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5. Multipliers on bi-parameter Haar system Hardy spaces

   https://doi.org/10.1007/s00208-024-02887-9  

   Lechner, R.; Motakis, P.; Müller, P_F_X; Schlumprecht, Th (May 2024, Mathematische Annalen)

   Abstract Let$$(h_I)$$ $\left(h_I\right)$denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$ $I\in D$, the set of dyadic intervals and$$h_I\otimes h_J$$ $h_I\otimes h_J$denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto h_I\left(s\right)h_J\left(t\right)$,$$I,J\in \mathcal {D}$$ $I,J\in D$. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$ $V\left({\delta }^2\right)$of$$h_I\otimes h_J$$ $h_I\otimes h_J$,$$I,J\in \mathcal {D}$$ $I,J\in D$. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$ $L^p[0,1]$or the Hardy spaces$$H^p[0,1]$$ $H^p[0,1]$,$$1\le p < \infty $$ $1\le p<\infty$. We say that$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D(h_I\otimes h_J)=d_{I,J}{h}_I\otimes h_J$, where$$d_{I,J}\in \mathbb {R}$$ $d_{I,J}\in R$, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta }^2\right)\to V\left({\delta }^2\right)$given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_I\otimes h_J=h_I\otimes h_J$if$$|I|\le |J|$$ $\left|I\right|\le \left|J\right|$, and$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_I\otimes h_J=0$if$$|I| > |J|$$ $\left|I\right|>\left|J\right|$, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$, there exist$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\text{Id}-C)\text{approximately 1-projectionally factors through}D,\end{array}$i.e., for all$$\eta > 0$$ $\eta >0$, there exist bounded operatorsA, Bso thatABis the identity operator$${{\,\textrm{Id}\,}}$$ $\text{Id}$,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert ·\Vert B\Vert =1$and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\text{Id}-C)-ADB\Vert <\eta$. Additionally, if$$\mathcal {C}$$ $C$is unbounded onX(Y), then$$\lambda = \mu $$ $\lambda =\mu$and then$${{\,\textrm{Id}\,}}$$ $\text{Id}$either factors throughDor$${{\,\textrm{Id}\,}}-D$$ $\text{Id}-D$. 

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