Computability of probability measures and Martin-Lof randomness over metric spaces

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).Comment: 29 page

Attacks on the Search-RLWE problem with small errors

The Ring Learning-With-Errors (RLWE) problem shows great promise for post-quantum cryptography and homomorphic encryption. We describe a new attack on the non-dual search RLWE problem with small error widths, using ring homomorphisms to finite fields and the chi-squared statistical test. In particular, we identify a "subfield vulnerability" (Section 5.2) and give a new attack which finds this vulnerability by mapping to a finite field extension and detecting non-uniformity with respect to the number of elements in the subfield. We use this attack to give examples of vulnerable RLWE instances in Galois number fields. We also extend the well-known search-to-decision reduction result to Galois fields with any unramified prime modulus q, regardless of the residue degree f of q, and we use this in our attacks. The time complexity of our attack is O(nq2f), where n is the degree of K and f is the residue degree of q in K. We also show an attack on the non-dual (resp. dual) RLWE problem with narrow error distributions in prime cyclotomic rings when the modulus is a ramified prime (resp. any integer). We demonstrate the attacks in practice by finding many vulnerable instances and successfully attacking them. We include the code for all attacks

The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$f(a):= \widetilde{\lim}_{m \to \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_0(n+ah_0) \dots g_k(n+ah_k)}{n}$$ where $1 \leq \omega_m \leq x_m$ are numbers going to infinity as $m \to \infty$, and $\widetilde{\lim}$ is a generalised limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely that these sequences $f$ are the uniform limit of periodic sequences $f_i$. Furthermore, if the multiplicative function $g_0 \dots g_k$ "weakly pretends" to be a Dirichlet character $\chi$, the periodic functions $f_i$ can be chosen to be $\chi$-isotypic in the sense that $f_i(ab) = f_i(a) \chi(b)$ whenever $b$ is coprime to the periods of $f_i$ and $\chi$, while if $g_0 \dots g_k$ does not weakly pretend to be any Dirichlet character, then $f$ must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three, and of the M\"obius function of length up to four.Comment: 41 pages, no figures. Submitted, Duke Math. J.. Referee changes incorporate

Multiple concentric annuli for characterizing spatially nonuniform backgrounds

A method is presented for estimating the background at a given location on a sky map by interpolating the estimated background from a set of concentric annuli which surround this location. If the background is nonuniform but smoothly varying, this method provides a more accurate (though less precise) estimate than can be obtained with a single annulus. Several applications of multi-annulus background estimation are discussed, including direct testing for point sources in the presence of a nonuniform background, the generation of "surrogate maps" for characterizing false alarm rates, and precise testing of the null hypothesis that the background is uniform.Comment: 35 pages, including 19 embedded postscript figures; LaTeX with AAS macros. Minor revisions, improved figures, as suggested by referee. To appear in Astrophysical Journa

Connected Choice and the Brouwer Fixed Point Theorem

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak K\H{o}nig's Lemma. While we can present two independent proofs for dimension three and upwards that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upwards. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.Comment: 36 page