On Sums of Generating Sets in ℤ₂ⁿ

Abstract

Let A and B be two affinely generating sets of Z n 2 . As usual, we denote their Minkowski sum by A + B. How small can A + B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of Z n 2 . These cosets are arranged as Hamming balls, the smaller of which has radius 1.

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“…A sequence of papers [25,62,39,45,29] established this conjecture for torsion r = 2, (i. e., to the groups G = F n 2 ) with C = 2, and Even-Zohar and Lovett [30] extended it to any prime torsion, also with C = 2. Theorem 2.9 (Even-Zohar and Lovett).…”

The Freiman–Ruzsa theorem over finite fields

“…For p = 2 and v = e i , the compression operator C v coincides with C {i} as in [8,6]. Compressions along general multidimensional subspaces can be defined analogously, but are not necessary for this work.…”

“…In particular, Green and Tao [8] showed that F (2, K) ≤ 2 2K+O( √ K log K) , thus settling Conjecture 2 for r = 2. A refinement of their argument enabled the first author [6] to find the exact value of F (2, K), which turned out to be Θ(2 2K /K). In this note we extend these techniques to the case of general prime torsion.…”

“…The proof elaborates on methods of subset compressions in F n 2 , which were first employed in the present context by Green and Tao [8]. Although the general scheme of the proof is similar to [6], the transition from 2 to general p requires a new type of compressions. The definition and properties of these compressions appear in Section 2.…”

“…In the special case r = 2, further progress has been made [4,12,8,10,6]. In particular, Green and Tao [8] showed that F (2, K) ≤ 2 2K+O( √ K log K) , thus settling Conjecture 2 for r = 2.…”