Large sum-free sets in ternary spaces

Abstract

We show that any sum-free subset A ⊆ F 3 r (with an integer r ⩾ 3 ), satisfying | A | > 5 · 3 r - 3 , is contained in a hyperplane. This bound is best possible: there exist sum-free subsets of cardinality | A | = 5 · 3 r - 3 not contained in a hyperplane. Conjecturally, if A ⊆ F 3 r is maximal sum-free and aperiodic (not a union of cosets of a non-zero subgroup), then | A | ⩽ ( 3 r - 1 + 1 ) / 2 . If true, this is best possible and allows one for any fixed ε > 0 to establish the structure of all sum-free subsets A ⊆ F 3 r such that | A | > ( 1 / 6 + ε ) · 3 r .