Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

We study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

Constructing sequences one step at a time

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain ([Formula: see text]) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma ([Formula: see text]), including showing that [Formula: see text] does not imply the thin set theorem for pairs, and that the principle “the product of well-quasi-orders is a well-quasi-order” is strictly between [Formula: see text] and the Ascending/Descending Sequences principle, even in the presence of [Formula: see text].

On notions of computability-theoretic reduction between Π21 principles

Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer (2016) on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa (to appear). Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman (2001).

Criteria of Wiener Type for Minimally Thin Sets and Rarefied Sets Associated with the Stationary Schrödinger Operator in a Cone

We give some criteria fora-minimally thin sets anda-rarefied sets associated with the stationary Schrödinger operator at a fixed Martin boundary point or ∞ with respect to a cone. Moreover, we show that a positive superfunction on a cone behaves regularly outside ana-rarefied set. Finally we illustrate the relation between thea-minimally thin set and thea-rarefied set in a cone.

Expected coalescence time for a nonuniform allocation process

We study a process where balls are repeatedly thrown into n boxes independently according to some probability distribution p. We start with n balls, and at each step, all balls landing in the same box are fused into a single ball; the process terminates when there is only one ball left (coalescence). Let c := ∑jpj2, the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c−1, under two constraints on p that exclude a thin set of distributions p. One of the constraints is c = o(ln−2n). This ln−2n is shown to be a threshold value: for c = ω(ln−2n), there exists p with c(p) = c such that the expected coalescence time far exceeds c−1. Connections to coalescent processes in population biology and theoretical computer science are discussed.

Expected coalescence time for a nonuniform allocation process

We study a process where balls are repeatedly thrown into n boxes independently according to some probability distribution p . We start with n balls, and at each step, all balls landing in the same box are fused into a single ball; the process terminates when there is only one ball left (coalescence). Let c := ∑ j p j 2, the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c −1, under two constraints on p that exclude a thin set of distributions p . One of the constraints is c = o(ln−2 n). This ln−2 n is shown to be a threshold value: for c = ω(ln−2 n), there exists p with c( p ) = c such that the expected coalescence time far exceeds c −1. Connections to coalescent processes in population biology and theoretical computer science are discussed.

The number of imaginary quadratic fields with a given class number

International audience This paper formulates some conjectures for the number of imaginary quadratic fields of a given class number. It establishes an asymptotic formula for the number of such fields with class number below $H$, and also shows that many fields have class numbers lying outside a very thin set.