@1FastCasper (and others).

I've recopied the office BMW AG tire size for the E53, as they've done the math on tire sizes. You'll see the wheel size as well. This means that when they do your technical inspection in Germany, Switzerland (where I live) they check this.

On a modern BMW, failure to have tire circumference within a roughly 2% tolerance means that the ASC and other systems think that the incorrect sized axle is turning too fast or too slow relative to the correct one - which in normal instances would indicate wheel spin or similar, so it flashes the light and tries to assist (as it's designed this way). The comment about stress on the transfer case is correct, the imbalance of tire diameter on a 4 wheel drive car adds a physical issue of wear imbalance on top of confusing your car's electronics.

Your issue on no problems on your 2001 is logical, the error in diameter wasn't enough to bother your ASC system and a 2001 doesn't have iDrive (my E53 is a 2002). iDrive is less tolerant, so not having (see chart below) 315/35-20 on the rear axle doesn't make the electronics see the correct number of tire rotations (relative to the front with the correct 275/40-20).

I had a colleague who put the wrong aspect ratio tires on his early 2000's E46 M3 and the lights were on all of the time. Put the correct tires on the car, problem resolved.

I'll leave wheel offset out, but this is equally important.

235/65-17, wheel size is 7.5 x 17" (same front & rear)

255/55-18", wheel size is 8.5 x 18" (same size front & rear)

255/50-19", wheel size is 9 x 19" (front)
285/45-19", wheel size is 10 x 10" (rear)

275/40-20", wheel size is 9.5 x 20" (front)
315/35-20", wheel size is 10.5 x 20" (rear)

If you want to check BMW's math, there is a handy Excel tire & wheel offset calculator done by a Canadian fellow, Aaron ?. Surf and you'll find it.

If putting the right wheel & tire size on your car doesn't do it, and/or cleaning your wheel speed sensors and cog ring (very good suggestion) doesn't resolve this - it would be handy to read out the body fault codes at a dealer or a good independent shop.

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2002 Euro X5 (E53) 3.0 diesel (M57)
5 speed manual with UUC short shifter.
Pre-xDrive NV125 transfer case (chain & gears).
Factory Xenon headlights, Spyder LED tail lights
Updated vortex crankcase breather.
Motorsport dead pedal, factory fire extinguisher, factory first aid kit, factory F&R mud flaps, factory PDC

Bilstein B6 shocks F&R
Redline oil has replaced "lifetime" fluids (F&R diff, manual transmission, transfer case and P/S).
Dimple magnetic drain plugs in all compartments.