Parametric Additive Number Theory: With an insight to Goldbach's Conjecture

In this monograph we present sufficient conditions to know if given a positive real number x = 2s and a uniformly discrete set of positive real numbers P, such a number is close enough to the sum of two elements of this set when s does not belong to P. In fact these conditions allows us to obtain bounds for the distance between x = 2s and the sum set P + P in a constructive way. The properties of the distribution function of the set P, F, play an essential role, as expected. We study the case consist that the distribution function of P is subadditive, what is connected to the set of the prime numbers through the Second Hardy-Littlewood Conjecture. Moreover, we also study the wider case consisting of F being relatively subadditive, what is verified by the distribution function of prime numbers (proved by Pierre Dusart in his doctoral thesis). We also distinguish the cases when the distribution function is relatively contractive and when it is not, and obtain results in both cases, using the distribution function in order to estimate distances to P + P.

Given a positive real number x = 2s and a uniformly discrete set of positive real numbers P, we investigate sufficient conditions to determine bounds of the distance between such a number and the sums of two elements of this set. We obtain several constructive results which allows us to know these bounds and approximations to the sum set P + P. First we obtain an upper approximation of x = 2s by a sum of two numbers of P, and then we adjust and improve this approximation using the properties of the distribution function consisting of being absolutely or relatively subadditive and also using the eccentricity of the half of the given number, and distinguising between distribution functions which are relatively contractive and those which are not. From other point of view we obtain estimates for the elements of P + P. We use upper bounds for the distribution function in order to obtain results of approximation. We also study these results in the context of the prime numbers.

In Chapter 9 we obtain new conditions to determine if given a real number and a uniformly discrete set of real numbers, such a number can be expressed as sum of two elements of this set. Although we obtain several general results, our work is motivated by the particular case of the prime numbers set. Namely, we establish a relationship between bounds for the distribution function such as that of the Second Hardy-Littlewood Conjecture and the Goldbach's Property for uniformly discrete sequences of real numbers.

Parametric Number Theory is the part of Number Theory which studies the distribution of uniformly discrete sets of real numbers when their distribution function is completely known except for one or several real parameters, what allows us to obtain some information on the distribution of such sets, and, of course, results on the distribution of uniformly discrete sets whose distribution functions belong to a same family of distributions.