arXiv subjects
Publications and source records attributed to Fan Chang.
Fang and Huang proved that every cancellative pair $(\mathcal{A},\mathcal{B})$ of families of subsets of $[n]$ satisfies $|\mathcal{A}|\cdot|\mathcal{B}|\leq(\frac{9}{4})^n$. Their proof uses entropy. We give a Fourier-analytic proof based on Lifshitz's one-sided noise operator $\mathrm{T}^{1/4\to1/2}$. The main tool is a near-$L^1$ hypercontractive estimate for this operator.
Frankston, Kahn and Narayanan proved that every regular increasing $3$-wise intersecting family of subsets of $[n]$ has cardinality $o(2^n)$ using Friedgut's junta theorem. We give a short quantitative proof using elementary tools from the analysis of Boolean functions and entropy. More precisely, if $\mathcal{A}\subseteq\mathcal{P}_n$ is a nonempty $3$-wise intersecting family that is both regular and increasing, then $$ \log\frac{2^n}{|\mathcal{A}|}\ge \frac{n}{2}\left(\frac{|\mathcal{A}|}{2^n-|\mathcal{A}|}\right)^2, $$ and consequently $|\mathcal{A}|\le 2^n\sqrt{W(n)/n}$, where $W$ is the principal Lambert function defined by $W(x)e^{W(x)}=x$ for $x\ge0$. We also give a purely Fourier-analytic proof of the weaker estimate $$ |\mathcal{A}|\le \frac{2^n}{1+n^{1/3}}. $$
Let $\mathcal{F}\subseteq\binom{[n]}k$, and let $\partial_{k\to\ell}\mathcal{F}$ be the family of all $\ell$-sets containing a member of $\mathcal{F}$. Writing $\mu_k(\mathcal{F})=|\mathcal{F}|/\binom nk$, $p=k/n$, and $q=\ell/n$, we prove $$ \mu_\ell(\partial_{k\to\ell}\mathcal{F})\geq \begin{cases}\mu_k(\mathcal{F})^{\frac{\log q}{\log p}}, &0\leq \mu_k(\mathcal{F})\leq p^2,\\ q\left[1-\left(1-\frac{\mu_k(\mathcal{F})}{p}\right)^{ \frac{\log(1-q)}{\log(1-p)}} \ \ \right], &p^2\leq \mu_k(\mathcal{F})\leq p,\\ 1-(1-\mu_k(\mathcal{F}))^{\frac{\log(1-q)}{\log(1-p)}}, &p\leq \mu_k(\mathcal{F})\leq1. \end{cases} $$ This yields a dimension-free closed-form lower bound for the finite Kruskal--Katona profile and each branch is asymptotically sharp. As the main application, we settle both the uniform and biased product conjectures of Frankl and Tokushige. If $r\geq2$, $0\leq k_i\leq(r-1)n/r$, and $\mathcal{F}_i\subseteq\binom{[n]}{k_i}$ are $r$-cross-intersecting, then $$ \prod_{i=1}^r\mu_{k_i}(\mathcal{F}_i)\leq\prod_{i=1}^r\frac{k_i}{n}. $$ In biased product setting, we prove that for $r$-cross-intersecting families $\mathcal{A}_i\subseteq 2^{[n]}$ and $0\leq p_i\leq(r-1)/r$, $\prod_{i=1}^r\mu_{p_i}(\mathcal{A}_i)\leq\prod_{i=1}^rp_i.$ We further determine all equality cases in both settings.
We prove the sharp diagonal spectral correlation conjecture of Friedgut, Kahn, Kalai and Keller, proposed in their Fourier-analytic approach to Chv\'atal's conjecture. For every pair of increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$, $$\mathrm{Cov}(f,g)\ge4\sum_{\varnothing\ne S\subseteq[n]}|S|\hat{f}(S)^2\hat{g}(S)^2.$$ Thus covariance controls the degree-weighted collision of the two nonconstant Fourier spectra, giving a sharp Fourier strengthening of the Harris--Kleitman inequality. The theorem also implies the unweighted diagonal conjecture of Friedgut--Kahn--Kalai--Keller for an increasing family and a maximal intersecting family. The factor $4$ is optimal, and we determine all equality cases. Apart from pairs whose relevant coordinate sets are disjoint, equality occurs only for a common dictatorship and, up to relabelling coordinates and interchanging $f$ and $g$, for the two-coordinate AND-OR pair $(f,g)=(x_i x_j,\,x_i\vee x_j).$ The main novelty is a correlated four-restriction induction and a sharp endpoint convolution inequality. The usual two-restriction induction behind Harris--Kleitman sees only the parallel restricted pairs and loses the mixed Fourier information needed to control the degree-weighted diagonal spectral energy. We instead couple the four codimension-one restricted pairs with correlation $1/2$; this precise correlation extracts the missing degree-weighted energy as a nonnegative square.
We prove a sharp product theorem for $r$-cross-intersecting families in the $p$-biased measure. If $r\ge2$, $0\le p\le \frac{r-1}{r}$, and $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq 2^{[n]}$ are $r$-cross-intersecting, then $$\prod_{i=1}^r \mu_p(\mathcal{F}_i)\le p^r.$$ The bound is attained by a common $1$-star, and the range of $p$ is best possible. In particular, this proves the equal-bias case of a conjecture of Frankl and Tokushige and, for $r=3$, confirms a conjecture of Tokushige. We also prove a stability theorem: for $r\ge3$, every near-extremal $r$-tuple is close, in $p$-biased measure, to a common $1$-star, with an optimal linear dependence on the product deficit. The extremal proof uses a coordinatewise coupling at the critical bias together with an isoperimetric inequality for increasing families. The stability proof uses dual families and random ordered partitions to obtain Fourier concentration, then applies biased Friedgut--Kalai--Naor theorem to force the star structure.
Talagrand's correlation inequality provides a quantitative strengthening of the Harris--Kleitman inequality for increasing Boolean functions. Motivated by a Fourier-analytic conjecture of Friedgut, Kahn, Kalai, and Keller, we prove that $$ \mathrm{Cov}(f,g)\ge 2\sum_{S\neq\emptyset}|S|\hat f(S)^2\hat g(S)^2 $$ holds for all increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$. The proof combines the reverse Bonami--Beckner inequality with Young's convolution inequality. We also establish a sharp pointwise inequality: for every $n\ge1$, every $0\le\rho\le1$, and every $f,g:\{0,1\}^n\to[0,1]$, the optimal constant $c_{\rho,n}$ for which $$ \left\langle f,T_\rho g \right\rangle\ge c_{\rho,n}\|f*g\|_2^2 $$ holds for all such $f,g$ is $1$ for $0\le\rho\le1/2$, $(2(1-\rho))^n$ for $1/2<\rho<1$, and $0$ for $\rho=1$. Integrating this pointwise inequality yields, for $n\ge1$, the slightly improved bound $$ \mathrm{Cov}(f,g)\ge 4\cdot\frac{n+1}{2n}\sum_{S\neq\emptyset}|S|\hat f(S)^2\hat g(S)^2. $$
Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the weaker bound \[ \mathbf{BSA}[f]\le \exp(C_d\sqrt{\log n}). \]
We establish dimension-free quantum Talagrand-type inequalities with explicit constants on the quantum Boolean cube, via a unified variance-decay perspective. For individual observables, short-time variance decay along the depolarizing semigroup, with rates estimated through hypercontractivity, naturally yields Talagrand-type bounds. Within this framework, we derive Talagrand-type energy--variance and high-order influence--variance inequalities. From the former, we obtain quantum analogues of Talagrand's isoperimetric inequality, the Eldan--Gross inequality, and the Cordero-Erausquin--Eskenazis inequality; from the latter, we derive high-order quantum Talagrand--KKL-type and partial isoperimetric bounds. Altogether, our work provides a variance-decay framework of broad applicability, unifying first-order and high-order Talagrand-type phenomena.
We study Boolean functions on the $p$-biased hypercube $(\{0,1\}^n,\mu_p^n)$ through the lens of Fourier (spectral) entropy, i.e. the Shannon entropy of the squared $p$-biased Fourier coefficients. Motivated by recent restriction-based advances on upper bounds toward the Fourier-Entropy-Influence (FEI) conjecture, we prove a complementary, sharp lower bound that decomposes the entropy into coordinate-wise contributions. Let $q:=4p(1-p)$ and define $\Psi:[0,\tfrac12]\to[0,\ln 2]$ by $\Psi(t):=h\left(\frac{1+\sqrt{1-4t^2}}{2}\right)$, where $h(u):=-u\ln u-(1-u)\ln(1-u)$. We show that for every Boolean $f:(\{0,1\}^n,\mu_p^n)\to\{\pm1\}$, $$ \mathrm{Ent}_p(f) \ge \sum_{k=1}^n \Psi\left(\sqrt{q(1-q)}\cdot\mathrm{Inf}_k^{(p)}[f]\right). $$ When $p\neq \tfrac12$, this bound is tight and equality holds if and only if $f$ is a parity function. Our proof adapts the restriction-moment framework to the biased cube.
Talagran's correlation inequality provides quantitative lower bounds on the covariance of two increasing Boolean functions in terms of their coordinate influences, but, in general, a logarithmic loss is necessary. Motivated by a question of Kalai, Keller and Mossel, we identify a natural log-free regime. We prove that if two increasing Boolean functions on $\{0,1\}^n$ are either both submodular or both supermodular, then $$ \mathbb{E}[fg]-\mathbb{E}[f]\mathbb{E}[g]\ge \frac{1}{4}\cdot\sum\limits_{i=1}^n\mathrm{Inf}_i[f]\mathrm{Inf}_i[g], $$ where the constant $1/4$ is optimal. We also prove a real-valued extension: for two functions with the same second-difference sign, the covariance is bounded below by the sum of products of their Level-1 Fourier coefficients. As a consequence, we verify the Friedgut--Kahn--Kalai--Keller spectral conjecture in this structured setting. The proofs combine a heat-semigroup representation based on second-order discrete derivatives with an independent induction argument for the Boolean case.
We prove a support--shattering uncertainty principle for functions on the Boolean cube. Let $\mathbb{F}$ be any field and let $f:\{0,1\}^n\to\mathbb{F}$ be nonzero. If $x^S$ is a maximum-degree monomial in the multilinear representation of $f$, then $\mathrm{supp}(f)$ shatters $S^c$. Consequently, \[ \mathrm{VC}(\mathrm{supp}(f))+\mathrm{deg}_{\mathbb{F}}(f)\ge n . \] For Boolean-valued functions this yields both the real-degree and algebraic-degree forms of the inequality. We derive several consequences. The real Fourier support of a nonzero Boolean function satisfies \[ \mathrm{VC}(\mathrm{Spec}(f))\ge \mathrm{deg}_{\mathbb{F}_2}(f), \qquad \mathrm{VC}(\mathrm{supp}(f))+\mathrm{VC}(\mathrm{Spec}(f))\ge n . \] The same principle gives a low-degree-obstruction proof of the Sauer--Shelah lemma and a product-space analogue based on the Efron--Stein decomposition, recovering the Karpovsky--Milman multivalued Sauer lemma. We also obtain polynomial-method applications, including an arbitrary-field Sziklai--Weiner lower bound and a shattering theorem for null designs, and discuss sharpness and equality examples.
Motivated by random walks on subsets of the hypercube, we prove two discrete functional inequalities on the hypercube by the technique of induction-by-restrictions. First, we give a short, elementary proof of the Poincar\'e inequality on increasing subsets of the cube recently established by Fei and Ferreira Pinto Jr, which yields an $O(n^2)$ upper bound on the mixing time of censored random walks, improving upon previous bounds. Second, adapting Samorodnitsky's induction method to the $p$-biased setting, we establish a sharp $p$-biased edge-isoperimetric inequality for real-valued functions supported on increasing sets, which recovers the classic biased edge-isoperimetric inequality for increasing sets and identifies increasing subcubes as the extremizers. This result also admits a probabilistic interpretation in terms of maximizing the mean first exit time of biased random walks known as Glauber dynamics.
The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $\delta(G) \ge \max \{ \frac{1}{2},\frac{r - \ell}{r} \}n + o(n)$ and $\alpha_{\ell}(G) = o(n) $ contains a $K_{r}$-factor? We give a negative answer for the case when $\ell\ge \frac{3r}{4}$ by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers $r,\ell$ with $r > \ell \ge \frac{3}{4}r$ and $\mu >0$, there exist $\alpha > 0$ and $N$ such that for every $n\in r\mathbb{N}$ with $n>N$, every $n$-vertex graph $G$ with $\delta(G) \ge \left( \frac{1}{2-\varrho_{\ell}(r-1)} + \mu \right)n $ and $\alpha_{\ell}(G) \le \alpha n$ contains a $K_{r}$-factor. Here $\varrho_{\ell}(r-1)$ is the Ramsey--Tur\'an density for $K_{r-1}$ under the $\ell$-independence number condition.