Carries, Group Theory, and Additive Combinatorics

Combinatorics of balanced carries

“…Carries and cocycles make sense for any subgroup H of any group G. Choosing coset representatives X for H in G and then picking elements x in X from some natural probability distribution leads to a carries process. This is developed in [4] and [12]. Developing a parallel theory involving a nested sequence of subgroups (as in the present paper) suggests a world of math to be done.…”

“…For general b, balanced carries lead to (b 2 − 1)/4 carries. This is the smallest number possible [2], [12].…”

Minimizing the Number of Carries in Addition

“…For every > 0 there exists a number p 0 = p 0 ( ) so that for any prime p > p 0 the probability of carry when adding two random independent 1-digit numbers using any fixed set of digits in base p is at least 1 4 − . The estimate given in [5] to p 0 = p 0 ( ) is a tower function of 1/ . Here we establish a tight result for any prime p, proving the conjecture for any prime.…”

“…• After the completion of this note I learned from the authors of [5] that closely related results (for addition in Z p , not in Z p 2 ) appear in the paper of Lev [6].…”

A robust version of Freiman's 3k–4 Theorem and applications

“…As an application to Theorem 1.1, we prove a structure theorem for sets A with the quantity C(A) := #{(a, a ′ ) ∈ A × A : a + a ′ ∈ A} |A| 2 being close to its maximum value 3/4 + o (1). The fact that C(A) is maximized when A is a symmetric interval around the origin can be proved by a simple combinatorial argument; see [5,Proposition 2.1]. The following result asserts that if C(A) is close to 3/4, then A must be close to a symmetric arithmetic progression around the origin.…”