What causes color differences in beach sand?
The colors of beaches are directly related to the sources of the sand. The darker sands contain more dark minerals, the very light, white sands are typically macerated calcareous material from reefs or shells. The purple you have seen may well be manganepidote or piedmontite, often associated with glaucophane-bearing rocks in the crystalline-schist. These shists can be found in the mountains to the northwest of Long Island. The rock which contains piedmontite as an essential component is well characterized in outward appearance, being of a dark violet colour. In your case on Long Island, the island itself is a remnant of a glacial terminal moraine which has moved rock and sediments from the northwest to Long Island Sound. So, the various minerals transported by the glacial activity can leave a variety of colors of sands from the varielty of rocks the glaciers abraded on the way to Long Island Sound.