On Extensions of the Steinhaus Theorem for Distance Sets and Difference Sets

Some Results on Sets of Positive Measure in a Metric Space

“…Now we present some definitions which may be readily seen in Boardman [2] and Lahiri [3]. Definition 1.…”

“…, which may be deduced from Theorem 13.5 of Munroe [4] as indicated by Boardman [2]. Let δ be such that…”

“…E. Boardman [2] proved that under certain conditions on the metric space (E, ) all simple rectifiable curves in E have the Steinhaus property for distance sets.…”