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1. 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 (February 2023, 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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2. 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 (April 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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3. 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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4. Universality and phase transitions in low moments of secular coefficients of critical holomorphic multiplicative chaos

   https://doi.org/10.1007/s00440-025-01443-z  

   Gu, Haotian; Zhang, Zhenyuan (November 2025, Probability Theory and Related Fields)

   Abstract We investigate the low moments$$\mathbb {E}[|A_N|^{2q}],\, 0 $E\left[\right|A_N{|}^{2q}],0<q⩽1$of secular coefficients$$A_N$$ $A_N$of the critical non-Gaussian holomorphic multiplicative chaos, i.e. coefficients of$$z^N$$ $z^N$in the power series expansion of$$\exp (\sum _{k=1}^\infty X_kz^k/\sqrt{k})$$ $exp({\sum }_{k=1}^{\infty }{X}_k{z}^k/\sqrt{k})$, where$$\{X_k\}_{k\geqslant 1}$$ ${\left\{X_k\right\}}_{k⩾1}$are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper’s remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each$$X_k$$ $X_k$is standard complex Gaussian,$$A_N$$ $A_N$features better-than-square-root cancellation:$$\mathbb {E}[|A_N|^2]=1$$ $E\left[\right|A_N{|}^2]=1$and$$\mathbb {E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$$ $E\left[\right|A_N{|}^{2q}]\asymp {(logN)}^{-q/2}$for fixed$$q\in (0,1)$$ $q\in (0,1)$as$$N\rightarrow \infty $$ $N\to \infty$. We show that this asymptotics holds universally if$$\mathbb {E}[e^{\gamma |X_k|}]<\infty $$ $E\left[e^{\gamma |X_k|}\right]<\infty$for some$$\gamma >2q$$ $\gamma >2q$. As a consequence, we establish the universality for the tightness of the normalized secular coefficients$$A_N(\log (1+N))^{1/4}$$ $A_N{(log(1+N))}^{1/4}$, generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of$$\mathbb {E}[|A_N|^{2q}]$$ $E\left[\right|A_N{|}^{2q}]$for$$|X_k|$$ $|X_k|$following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper’s robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of$$A_N$$ $A_N$. 

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5. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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