Difference Balanced Functions and Their Generalized Difference Sets

Difference balanced functions from $F_{q^n}^*$ to $F_q$ are closely related to combinatorial designs and naturally define $p$-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the $d$-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be $d$-homogeneous. First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups. We then derive several necessary and sufficient conditions for $d$-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the $d$-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for $q$ prime. Furthermore, we show that every difference balanced function must be balanced or an affine shift of a balanced function.Comment: 17 page

Quasi-isometries Between Groups with Two-Ended Splittings

We construct `structure invariants' of a one-ended, finitely presented group that describe the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For groups satisfying two technical conditions, these invariants reduce the problem of quasi-isometry classification of such groups to the problem of relative quasi-isometry classification of the factors of their JSJ decompositions. The first condition is that their JSJ decompositions have two-ended cylinder stabilizers. The second is that every factor in their JSJ decompositions is either `relatively rigid' or `hanging'. Hyperbolic groups always satisfy the first condition, and it is an open question whether they always satisfy the second. The same methods also produce invariants that reduce the problem of classification of one-ended hyperbolic groups up to homeomorphism of their Gromov boundaries to the problem of classification of the factors of their JSJ decompositions up to relative boundary homeomorphism type.Comment: 61pages, 6 figure