Sums and differences of correlated random sets

Abstract

ABSTRACT. Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that |A − A| > |A + A| for a finite set A. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of {0, . . . , n} are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as n → ∞. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in A to go to zero, have been analyzed extensively.We consider many of these problems in a more general setting. Instead of just one set A, we study sums and differences of pairs of correlated sets (A, B). Specifically, we place each element a ∈ {0, . . . , n} in A with probability p, while a goes in B with probability ρ 1 if a ∈ A and probability ρ 2 if a ∈ A. If |A + B| > |(A − B) ∪ (B − A)|, we call the pair (A, B) a sum-dominant (p, ρ 1 , ρ 2 )-pair. We prove that for any fixed ρ = (p, ρ 1 , ρ 2 ) in (0, 1) 3 , (A, B) is a sum-dominant (p, ρ 1 , ρ 2 )-pair with positive probability, and show that this probability approaches a limit P ( ρ). Furthermore, we show that the limit function P ( ρ) is continuous. We also investigate what happens as p decays with n, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.

Problems in additive number theory, V: Affinely inequivalent MSTD sets

“…Steve Miller and his students and colleagues have contributed greatly to this subject (cf. [2,3,8,9,12,13,14,30,29,31]).…”

Geometry of Random Sparse Arrays

“…The most relevant body of work on difference sets of randomly generated sets can actually be found in the additive combinatorics literature. Many of these works nevertheless focus on comparing the size of the difference and sum sets of random sets for various sampling distribution [36]- [39]. An exception is Harvey-Arnold et al [40], who recently characterized the probability distribution of the number of missing elements in the difference set, when elements of the original set are chosen uniformly at random with a fixed probability.…”

Generalizing the Distribution of Missing Sums in Sumsets

“…As addition is commutative and subtraction is not, it was expected that in the limit almost all sets would be difference-dominant, though there were many constructions of infinite families of MSTD sets. 1 There is an extensive literature on such sets, their constructions, and generalizations to settings other than subsets of the integers; see for example [AMMS,BELM,CLMS,CMMXZ,DKMMW,He,HLM,ILMZ,Ma,MOS,MS,MPR,MV,Na1,Na2,PW,Ru1,Ru2,Ru3,Sp,Zh1].…”