Right now, I think Spurs are in the best position to take 4th, then Arsenal, then us, then West Ham.
West Ham has only 16 games left, of whom 6 is against the other top 7 teams, against whom they are usually fragile. We have 18 left, also with 6 against the other top 7 teams, and are 2 points behind West Ham. Arsenal, like us, have 18 left, but only 5 against the other top 7 teams, and they're 3 points ahead. They do however have a weak record against top teams. Spurs have a whopping 20 games remaining, and are 4 behind West Ham, 2 behind Arsenal, 1 ahead of us. Granted with 7 of those 20 games being against the other top 7 teams, but still.
Here's one way to look at it:
For reference, these are the average points per game taken by the top 6 teams on the table so far this season:
Man City: 2.5
Liverpool: 2.1
Chelsea: 2.0
West Ham: 1.7
Arsenal: 1.8
Spurs: 1.8
We took 1.42 under Ole (which is roughly similar to Brighton or Leicester now). Carrick raised the team average to 1.5, with a win and a draw in his two games. Under Rangnick it's been at 1.83, pushing the season average to 1.6.
To get to 70 points, we'll need an average of 2.11 for the rest of the season - or in other words, to perform roughly like Liverpool has so far. That would mean a very significant improvement on our performance under Rangnick so far, especially when you consider the opponents he's had and the opponents he will have.
Arsenal to get to the same points need 1.94, which is a bit better than their season average, but if you consider their horrific start they are probably already at more or less the required level, if they can keep it up.
Tottenham needs 1.85, which would only be a slight improvement on their season average. Since that average includes a period of falling apart and sacking the manager, that does not entail any need for signficant improvement on their current level.
West Ham needs 2.06, which is again Liverpool levels, and frankly seems well beyond realistic.
My guess would be that at least one of the other 3 teams will top 70 points this year, so that we'll need not much less than last year's tally (74) to make top 4.
The West Ham game on Saturday is huge.