avr_fred:
How are you powering the Nano? Sounds like not enough current, have you measured the supply voltage at the display?
I tried turning the segments on by displaying 8888 in a loop and they bright well. No problem with the power (and it's 5V at the display).
I solved the problem by altering the sketch. I count pulses in an interrupt to display rpm. There was a delay(1000) to display the reading every 1 second. Instead, I use a while() that exits after 1000 millis() and inside it I display the last reading, so the digits are always on.