What cable did you buy?
Most won't be bi-directional and are named SOURCE-DISPLAY, so you'd need HDMI to DVI-D, not DVI-D to HDMI.
Though generally true, SL-DVI DVI/HDMI is the exception to this, the adapters really are just wires, so DVI-HDMI (single-link) adapters will work both ways, and all DVI-HDMI adapters are single-link.
However
- If your monitor is Dual-Link DVI, you'd need an adapter.... Most are either passive (so not DL) or outright junk/fake.
- if your monitor actually has DVI-I (analog), you'd need an active adapter to convert the digital (HDMI) to analog (DVI-I)
Even if the monitor had dual-link DVI capability, it should still accept and work with single-link signals. Since 1680×1050 (presumably at 60 Hz) is well within the limits of SL-DVI it's unlikely the monitor supports dual-link signals anyway.
As for analog, all DVI ports are required to support digital DVI signals. DVI-I adds support for analog, but it's in addition to digital signals. There are no "analog-only" DVI ports on monitors, other than I think 1 or 2 models of CRT from 2000 that were the first on the market after the DVI specification was published, and manufacturers didn't know what they were doing yet.
By all accounts, it
should be working with the adapter, so I can only think that either the cable/adapter is bad (can you test it with another device?) or the PS4 might not support 1680×1050 (unlikely, since it should still be able to fall back to other standardized formats). Or maybe the monitor hasn't been changed to DVI input