This dumb phones-cause-fertility-decline theory is a kind of economics research I dislike. It's just like Melissa Kearney's old thing about 16 and Pregnant causing a decline in teen births. A reasonable hypothesis would be, "Birth rates are falling because fewer people are dating and marrying," ...
Ugh. Not good reporting about unrefereed preprints. Sabrina Tavernese at NYT says two studies were "published," one of which was "published in the National Bureau of Economic Research" (not a journal) the other is on SSRN (not identified).