Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n there exists an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball of equal volume embedded in (n+1)-dimensional Boolean cube, such that for all x and y it holds that distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y) where distance(,) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive. This result gives a strong negative answer to an open problem of Lovett and Viola [CC 2012], who raised the question in the context of sampling distributions in low-level complexity classes. The conceptual implication is that the problem of proving lower bounds in the context of sampling distributions will require some new ideas beyond the sensitivity-based structural results of Boppana [IPL 97]. We study the mapping f further and show that it (and its inverse) are computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that f is "approximately local" in the sense that all but the last output bit of f are essentially determined by a single input bit

Lipschitz bijections between boolean functions

We answer four questions from a recent paper of Rao and Shinkar on Lipschitz bijections between functions from $\{0,1\}^n$ to $\{0,1\}$. (1) We show that there is no $O(1)$-bi-Lipschitz bijection from $\mathrm{Dictator}$ to $\mathrm{XOR}$ such that each output bit depends on $O(1)$ input bits. (2) We give a construction for a mapping from $\mathrm{XOR}$ to $\mathrm{Majority}$ which has average stretch $O(\sqrt{n})$, matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $\phi : \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $\mathrm{XOR}(x) = \mathrm{Majority}(\phi(x))$ for all $x \in \{0,1\}^n$. (4) We show that with high probability there is a $O(1)$-bi-Lipschitz mapping from $\mathrm{Dictator}$ to a uniformly random balanced function

Biseparating maps on generalized Lipschitz spaces

Let $X, Y$ be complete metric spaces and $E, F$ be Banach spaces. A bijective linear operator from a space of $E$-valued functions on $X$ to a space of $F$-valued functions on $Y$ is said to be biseparating if $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are characterized as weighted composition operators, i.e., of the form $Tf(y) = S_y(f(h^{-1}(y))$ for a family of vector space isomorphisms $S_y: E \to F$ and a homeomorphism $h : X\to Y$. We also investigate the continuity of $T$ and related questions. Here the functions involved (as well as the metric spaces $X$ and $Y$) may be unbounded. Also, the arguments do not require the use of compactification of the spaces $X$ and $Y$

Evolution equations in non-commutative probability

The thesis presents two applications of evolution equations for non-commutative variables to the theory of non-commutative probability and von Neumann algebras.In the first part, non-commutative processes $(X_t)_{t \in [0,T]}$ with boolean, free, monotone, or anti-monotone independent increments, under certain continuity and boundedness assumptions, are classified in terms of certain evolution equations for their $F$-transforms $F_{X_t}(z) = (E[(z - X_t)^{-1}])^{-1}$. This classification is done in the setting of operator-valued non-commutative probability, in which the expectation takes values in a $\mathrm{C}^*$-algebra $\mathcal{B}$ rather than $\mathbb{C}$. Thus, the $F$-transform is a function of an operator variable $z$ from (matrices over) $\mathcal{B}$, and it is understood through the theory of fully matricial or non-commutative functions, an operator-valued analogue of complex analysis. The classification of these processes generalizes previous work on the L\'{e}vy-Hin\v{c}in formula for processes with independent and stationary increments, and it leads to Bercovici-Pata-type bijections between the processes with independent increments for the four different types of independence. We also describe a canonical model for each process with independent increments using operators on a Fock space. In fact, the interaction between operator models and analytic function theory is a major theme of the first part, and leads to a new ``coupling'' technique to prove estimates for the non-commutative central limit theorem and for Loewner chains.In the second part, we strengthen the probabilistic, information-theoretic, and transport-theoretic connections between asymptotic random matrix theory and tracial $\mathrm{W}^*$-algebras through the study of functions and differential equations for several non-commuting self-adjoint variables. We consider a random variable $X^{(n)}$ in $M_n(\mathbb{C})_{\text{sa}}^d$ given by a probability distribution$$d\mu^{(n)}(x) = \frac{1}{\int e^{-n^2 V^{(n)}}} e^{-n^2 V^{(n)}(x)}\,dx,$$where V^{(n)}: M_n(\C)_{sa}^d \to \R is uniformly convex and semi-concave. We assume that $(\nabla V^{(n)})_{n \in \N}$ is asymptotically approximable by trace polynomials, which means that $\nabla V^{(n)}$ behaves asymptotically like some element $f$ from a certain space of ``functions of $d$ self-adjoint variables from a tracial $\mathrm{W}^*$-algebra.''Then we show first that $X^{(n)}$ almost surely converges in non-commutative law to some $d$-tuple $X$ from a tracial $\mathrm{W}^*$-algebra $(\mathcal{M},\tau)$, meaning that (1/n) \Tr(p(X^{(n)})) \to \tau(p(X)) almost surely for every non-commutative polynomial $p$ (which is comparable to earlier known results). The strategy to prove convergence of the expectation E[(1/n) \Tr(p(X^{(n)}))] is to show that the heat semigroup associated to the measure $\mu^{(n)}$ preserves asymptotic approximability by trace polynomials. The same method leads to a new conditional version of this result, which shows that if k < d and if $(f^{(n)})$ is asymptotically approximable by polynomials, then so is the function $g^{(n)}$ given by $g^{(n)}(X_1^{(n)},\dots,X_k^{(n)}) = E[f^{(n)}(X^{(n)}) | X_1^{(n)}, \dots, X_k^{(n)}]$. Understanding the large-$n$ behavior of such conditional expectations is a key step in showing our second main result that the classical entropy of $X^{(n)}$, after renormalization, converges to Voiculescu's non-microstates free entropy $\chi^*(X)$ (and an analogous result for conditional entropy given $X_1^{(n)}$, \dots, $X_k^{(n)}$). In particular, we obtain a new proof of the result from a 2017 preprint of Dabrowski that $\chi^*(X)$ agrees with the microstates free entropy $\chi(X)$ for any $X$ that arises from such random matrix models.The final main result studies the large-$n$ behavior of certain functions $F^{(n)}$ that transport the measure $\mu^{(n)}$ to the distribution $\sigma_1^{(n)}$ of a standard Gaussian self-adjoint $d$-tuple $Z^{(n)}$. The transport map $F^{(n)}$ is obtained by the same construction as in Otto and Villani's famous proof of the Talagrand inequality based on heat semigroups and transport equations. Using successive conditioning, we can obtain a transport function $F^{(n)}$ that is lower-triangular in the sense that$$F^{(n)}(x_1,\dots,x_d) = (F_1^{(n)}(x_1), F_2^{(n)}(x_1,x_2),\dots,F_d^{(n)}(x_1,\dots,x_d)),$$where x = (x_1,\dots,x_d) \in M_n(\C)^d. We show that $F^{(n)}$ is asymptotically approximable by trace polynomials as $n \to \infty$, and consequently, in the large-$n$ limit, we obtain an isomorphism $\mathrm{W}^*(X_1,\dots,X_d) \to \mathrm{W}^*(Z_1,\dots,Z_d)$ that maps $\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(Z_1,\dots,Z_k)$ for every $k = 1, \dots, d$. As an application, we show that this statement holds when $X$ itself is given by $X_j = Z_j + \delta p_j(Z)$ where $Z$ is a free semicircular $d$-tuple, $p_j$ is a self-adjoint non-commutative polynomial, and $\delta$ is sufficiently small, depending on $p_1$, \dots, $p_d$

Trees of definable sets over the p-adics

To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the tree at depth i. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of Z_p^2, and for one-dimensional sets.Comment: 33 pages, 1 figur