I'm becoming more unhinged...
It's surprising that he uses a log when pursuing the Fellowship down the Anduin, as I assume then that he couldn't steal a boat nor build something while he waited for the Fellowship to leave Lothlorien.
It's just striking - at least today - that an addict like Gollum is in some way shown to be a builder of something. Maybe he had a little hut on his island, a little mushroom garden just outside, lined with a fishbone fence next to the shiny stone pathway that led down to the dock on his lake.