On extending Pollard’s theorem for t-representable sums

Abstract

Let t ≥ 1, let A and B be finite, nonempty subsets of an abelian group G, and let $A\mathop + \limits_i B$ denote all the elements c with at least i representations of the form c = a + b, with a ∈ A and b ∈ B. For |A|, |B| ≥ t, we show that either

$$ \sum\limits_{i = 1}^t {|A\mathop + \limits_i B| \geqslant t|A| + t|B| - 2t^2 + 1,} $$

((1))

or else there exist A′ ⫅ A and B′ ⫅ B with

$$ \begin{gathered} l: = |A\backslash A'| + |B\backslash B'| \leqslant t - 1, \hfill \\ A'\mathop + \limits_i B' = A' + B' = A\mathop + \limits_i B,and \hfill \\ \sum\limits_{i = 1}^t {|A\mathop + \limits_i B| \geqslant t|A| + t|B| - (t - l)(|H| - \rho ) - tl \geqslant t|A| + t|B| - t|H|,} \hfill \\ \end{gathered} $$

where H is the (nontrivial) stabilizer of $A\mathop + \limits_t B$ and

$$ \rho = |A' + H| - |A'| + |B' + H| - |B'|. $$

This gives a version of Pollard’s Theorem for general abelian groups in the tradition of Kneser’s Theorem. The proof makes use of additive energy and other recent advances in employing the Dyson transform. Two examples are given that show that such a Kneser-type result cannot hold when the bound in (1) is extended to the original bound of Pollard (for t ≥ 3), and that reduction present in (1) is of the correct order of magnitude (quadratic in t). However, in the case t = 2, we improve (1) to $|A\mathop + \limits_1 B| + |A\mathop + \limits_2 B| \geqslant 2|A| + 2|B| - 4$, which answers the abelian case of a question of Dicks and Ivanov related to extensions of the Hanna Neumann Conjecture.