It is proved that the induced squared-norm estimator is unbiased up to a term decaying geometrically with a sampling gap, and that its variance is a constant $V_0/m$ that is dimension-free in experiment and, under one stated concentration hypothesis, in theory.
Abstract
Zero-knowledge (ZK) proofs certify that a message belongs to an allowed semantic class without revealing the message, but the certificate compares a high-dimensional embedding against class centroids, so its cost grows with the embedding dimension $d$. A Johnson--Lindenstrauss (JL) projection lowers $d$ to $m\ll d$ while preserving pairwise distances, yet a random JL matrix must be committed and its sampling proved inside the circuit, which is costly and a leakage risk. We construct a public deterministic projection from the standardized orbit of a Pisot $\beta$-transformation, analyzed through the spectral gap of the $\beta$-map, the geometric decay of its correlations, rather than equidistribution. We prove that the induced squared-norm estimator is unbiased up to a term decaying geometrically with a sampling gap, and that its variance is $V_0/m$ with a constant $V_0$ that is dimension-free in experiment and, under one stated concentration hypothesis, in theory. A single public seed preserving all pairwise centroid distances therefore exists and is found by search. Against six standard projections, including the chaotic-sequence matrix of Yu \emph{et al.}, the construction matches statistical quality to within measurement noise, and it is the only one simultaneously free of in-circuit randomness and exactly reproducible in a fixed finite field at a per-step cost $\log_2\beta$ rather than $2^{k}$.
It is notoriously difficult to obtain deterministic reductions for the Minimum Distance Problem (MDP) and the Shortest Vector Problem (SVP). Under two-sided-error randomized reductions, Bennett, Cheraghchi, Guruswami, and Ribeiro (STOC 2023) proved parameterized hardness of approximation for these problems. We partially derandomize their reductions and present one-sided-error randomized reductions: MDP is W[1]-hard to approximate within an arbitrary constant factor under FPT many-one one-sided-error randomized reductions; For every $p \ge 1$, SVP in the $\ell_p$ norm is W[1]-hard to approximate within an arbitrary constant factor below $2^{1/p}$. We demonstrate the usefulness of one-sided-error randomized reductions by showing that they can be conditionally derandomized when the target problem has an OR function. Under a standard hardness-vs-randomness assumption, namely a plausible lower-bound assumption against nondeterministic circuits, we prove a general theorem formalizing this derandomization. Here, an OR function combines several instances into one instance that preserves their disjunction. We construct such OR functions for the relevant MDP and SVP gap problems, and thereby obtain deterministic W[1]-hardness for approximating MDP over every fixed finite field within every constant factor, and for approximating SVP in $\ell_p$ norms for every fixed integer $p$ within every factor below $2^{1/p}$. Applying the same framework to Micciancio's one-sided-error randomized reduction (ToC 2012) yields, under the same circuit lower-bound assumption, deterministic polynomial-time NP-hardness of approximating Euclidean SVP within every constant factor.
The conjectured upper bound of k statistical queries on a universe of size T can be released under pure differential privacy with expected worst-coordinate error at the square-root rate suggested by known lower bounds is proved.
Wirsching (2003) reduces uniform positive predecessor density for the $3n+1$ map to a chain of five conditions, organized into three conjectures. We prove two of them. Conjecture 1 concerns his path-counting generators: in the convolution carrying them to his Elka functions the partition weights grow subexponentially while the binomial ratio decays geometrically, so a window of width $O(\sqrt{\ell})$ dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density $\varphi$, a base-3 analogue of the Fabius density, against an explicit $\varphi_0$ due to Berg and Kr\"uppel. The exact log-Laplace transform of $\varphi$ splits into a smooth part, a log 3-periodic correction $H$, and a doubly exponentially small remainder. Berg and Kr\"uppel represented that correction as an infinite product in 1998; their analysis did not determine whether it is constant. We give $H$ as a Fourier series with coefficients in closed form in $\Gamma$ and $\zeta$; a classical zero-free theorem for $\zeta$ shows it is not constant, and we enclose its oscillation rigorously. Wirsching's comparison class fixes one phase of $H$, and there Conjecture 3 holds with limit $e^{H(0)}$, certified to lie in $(0.53412203666478,0.53412203666479)$. Off that class the phase sweeps a full period, so the unrestricted asymptotic $\varphi(t)\sim\kappa\varphi_0(t)$ fails. Condition $(\star4)$ follows, at every window radius, with $\mu=1/3$: what Wirsching's argument needs is weaker than Conjecture 3 itself, and the same saddlepoint chain settles it directly. With Conjecture 1 the chain reduces to the single condition $(\star3)$. Conjecture 2 is his route to it and remains open.
For a signing $\sigma$ of a $d$-regular graph, the spectrum of $A_\sigma$ depends only on the signs of cycles. We study the affine $\mathbb F_2$ family of signings making every short even cycle unbalanced, and show that averaging over it converts the sign problem of the Bilu-Linial conjecture into a counting problem: a master identity expresses the family-averaged trace as a parity-weighted sum over wrap classes confined to the span $W$ of the constraint cycles, and the family-averaged Ihara $L$-function diagonalizes so that every prime whose parity escapes $W$ contributes the Ramanujan rate $\sqrt{d-1}$ automatically. Uniform averaging over all signings, by contrast, provably cannot certify a spectral radius below the Kesten profile. We prove matched upper and lower bounds for the confined walk counts, a doubling injection from below, and from above an ear-decomposition encoding in which the number of fresh runs of a non-backtracking walk equals the cycle rank of its support, combined with a window lemma for bicycle-free graphs and a rank bound via the Moore bound for irregular graphs. Consequences include $\varepsilon$-versions of the Bilu-Linial conjecture: every $d$-regular graph that is subcritical at scale $\log n$, and every $d$-regular graph bicycle-free at radius $C\log\log n/\delta$, admits a signing in the parity family with $\rho(A_\sigma)\le2\sqrt{d-1}(1+C\delta\log(1/\delta))(1+o(1))$. We further identify the necessary hypotheses exactly ($K_d$-trapping; tree-burst gadgets), give an exact certificate on the hypercube, and record a decisive obstruction to two-sided interlacing: $\mathbb E_\sigma\det(xI-A_\sigma^2)$ is not real-rooted, already for the quadrilateral, where it equals $(x^2-4x+2)^2+4$.
We determine all equality cases in the Tu--Deng bound $|S_{t,k}|\le 2^{k-1}$. If the $k$-bit cyclic word of $t$ has $R$ ones, $Z$ zeros, and cyclic one-gap lengths $g_1,\ldots,g_Z$, then equality holds if and only if $g_i\ge Z-1$ for every $i$. This resolves Conjecture~3.20 of Flori, Randriambololona, Cohen and Mesnager, and we also enumerate all equality parameters. For $R\ge Z$ we determine the sharp first stability gap and all extremal words, while for $R<Z$ we obtain an exact quantization of the deficit and an explicit run-sensitive lower bound. The proofs are structural: an explicit matrix conjugation identifies the auxiliary enumerators in the two recent complete proofs of the Tu--Deng conjecture. We then develop a rooted coarsening model for all coefficients, prove one-sided deletion rigidity and an exact Macaulay-flux identity, and derive a Macaulay--M\"obius formula from the bounded simplex at the highest cyclic level.
We study random sketching matrices with Khatri-Rao structure. In particular, we consider the Khatri-Rao product (i.e., column-wise tensor product) $A_1\odot\cdots\odot A_d \in \mathbb R^{(n_1 \cdots n_d) \times m}$ of random matrices $A_i \in \mathbb R^{n_i \times m}$ whose columns are isotropic, independent and sub-Gaussian (e.g., Gaussian matrices). Khatri-Rao sketching matrices are widely applied in randomized algorithms for linear algebraic computation and data analysis, when the input data has tensor structure that allows for fast multiplication with $A_1\odot\cdots\odot A_d$. However, existing theory is not able to fully explain their performance in practice. In particular, despite significant attention, our best bounds for the important \emph{oblivious subspace embedding} property with Khatri-Rao matrices lag behind what is achievable with standard unstructured matrices. For embedding a $k$-dimensional subspace to $(1\pm \epsilon)$ error, Bujanovi\'c et al. \cite{bujanovic2025subspace} prove that sketching dimension $m = O(k^{3/2}/\epsilon^2)$ suffices in the special case of $d = 2$. Their dependence on $k$ is weaker than the tight bound of $O(k/\epsilon^2)$ known for unstructured sub-Gaussian sketching matrices. In this work, we close this gap, showing that $m = \tilde O(k/\epsilon^2)$ suffices for subspace embedding with a Khatri-Rao sketching matrix with any fixed order $d$. Our proof is simple, leveraging just two basic properties of the Khatri-Rao sketching distribution: 1) the columns of $A_1\odot\cdots\odot A_d \in \mathbb R^{(n_1 \cdots n_d) \times m}$ are independent and isotropic, and 2) each column of $A_1\odot\cdots\odot A_d \in \mathbb R^{(n_1 \cdots n_d) \times m}$ satisfies a weak Johnson-Lindenstrauss type moment property.
