Dust and shells and fish poop and all sorts of things fall to the sea bed to make sediment. Across broad central regions of the ocean, the sediment accumulates at a uniform rate-piling up about as rapidly here as it does over there. And, in most places, the currents don't move the sediment around much, so that it stays where it falls. Thus, the thickness of the sediment is related to the age of the rocks beneath the sediment. If you go around an ocean and measure the thickness of the sediment in lots of places, you are likely to find:

The sediment is thin near spreading ridges, and thicker away from the ridges.

Explanation:

Marine sediments are insoluble particles of rocks, soils, volcanoes, chemical and organic bodies that are carried from land to the ocean through streams and lakes. The particles are usually of different sizes, which are found either on the shores of the oceans or on the seabed in the form of small rocks and gravel. As stated in the question, in most places currents do not move the sediment very much so that it remains where it falls. However due to the constant movement of the current near the ridges, the sediments are constantly dragged, leaving a thin layer of sediment in this region. On the other hand, the sediment layer is thicker in the ridges region, because the current movement is less constant.