Chebotarev density theorem in short intervals for extensions of Fq(T)\mathbb{F}_q(T)

An old open problem in number theory is whether Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension $E$ of $\mathbb{Q}$ with Galois group $G$, a conjugacy class $C$ in $G$ and an $1\geq \varepsilon>0$, one wants to compute the asymptotic of the number of primes $x\leq p\leq x+x^{\varepsilon}$ with Frobenius conjugacy class in $E$ equal to $C$. The level of difficulty grows as $\varepsilon$ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime $1\geq\varepsilon>1/2$. We establish a function field analogue of Chebotarev theorem in short intervals for any $\varepsilon>0$. Our result is valid in the limit when the size of the finite field tends to $\infty$ and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem, and applied in a much more general setting of arithmetic functions, which we name $G$-factorization arithmetic functions.Comment: Incorporated referee comments. Accepted for publication in Trans. Amer. Math. So

MuLER: Detailed and Scalable Reference-based Evaluation

We propose a novel methodology (namely, MuLER) that transforms any reference-based evaluation metric for text generation, such as machine translation (MT) into a fine-grained analysis tool. Given a system and a metric, MuLER quantifies how much the chosen metric penalizes specific error types (e.g., errors in translating names of locations). MuLER thus enables a detailed error analysis which can lead to targeted improvement efforts for specific phenomena. We perform experiments in both synthetic and naturalistic settings to support MuLER's validity and showcase its usability in MT evaluation, and other tasks, such as summarization. Analyzing all submissions to WMT in 2014-2020, we find consistent trends. For example, nouns and verbs are among the most frequent POS tags. However, they are among the hardest to translate. Performance on most POS tags improves with overall system performance, but a few are not thus correlated (their identity changes from language to language). Preliminary experiments with summarization reveal similar trends

Chebotarev density theorem in short intervals for extensions of F_q(T)

An old open problem in number theory is whether the Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension E of Q with Galois group G, a conjugacy class C in G, and a 1 ≥ ε > 0, one wants to compute the asymptotic of the number of primes x ≤ p ≤ x+x^ε with Frobenius conjugacy class in E equal to C. The level of difficulty grows as ε becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ε > 1/2. We establish a function field analogue of the Chebotarev theorem in short intervals for any ε > 0. Our result is valid in the limit when the size of the finite field tends to ∞ and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem and applied in a much more general setting of arithmetic functions, which we name G-factorization arithmetic functions

Fine-Grained Analysis of Cross-Linguistic Syntactic Divergences

The patterns in which the syntax of different languages converges and diverges are often used to inform work on cross-lingual transfer. Nevertheless, little empirical work has been done on quantifying the prevalence of different syntactic divergences across language pairs. We propose a framework for extracting divergence patterns for any language pair from a parallel corpus, building on Universal Dependencies. We show that our framework provides a detailed picture of cross-language divergences, generalizes previous approaches, and lends itself to full automation. We further present a novel dataset, a manually word-aligned subset of the Parallel UD corpus in five languages, and use it to perform a detailed corpus study. We demonstrate the usefulness of the resulting analysis by showing that it can help account for performance patterns of a cross-lingual parser