Abstract: A result due to Hindman states that, no matter how the positive integers are finitely partitioned, one cell of the partition contains a sequence and all its sums without repetition. Straus, answering a question of Erdos, later gave an example showing that a density version of Hindman's result does not hold. He exhibited sets of positive integers with arbitrarily large density, each having the property that no shift contains a sums set of the above kind. In this talk I will present recent joint work with V. Bergelson, C. Christopherson and P. Zorin-Kranich in which we generalize Straus's example to a class of locally compact, second countable, amenable groups and show, using ergodic theory techniques recently developed by Host and Austin, that positive density subsets of groups outside this class must contains sets with strong combinatorial properties. In particular, this allows us to give a combinatorial characterization of minimally almost periodic, amenable groups.