Let A be a set of natural numbers. Recent work has suggested a strong link between the additive energy of A (the number of solutions to a 1 + a 2 = a 3 + a 4 with a i ∈ A) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of A modulo 1. There appears to be reasonable evidence to speculate a sharp Khintchinetype threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian. a,b∈A,a =b α(a−b) s/N 2010 Mathematics Subject Classification. 11J71, 11J83, 05B10, 11B30, 60F10.
Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound, i.e. ap(E) = ± 2 √ p . Assuming that all the symmetric power L-functions associated to E have analytic continuation for all s ∈ C, satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner [RT17], and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where ap(E) is fixed because of the Sato-Tate distribution.
We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert’s Irreducibility Theorem for degree $n$ polynomials $f$ with $\textrm {Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants.
We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree n polynomials f with Galpf q Ď An. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree n monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree n number fields with almost prime discriminants.
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