If we can find two points (p1 and p2) on opposite sides of the surface w = x3 + y3 − z3 = 0 we can look for intermediate points in the interval that provide better solutions. If we assume that the function is linear along the vector p1→p2 then the solution for w = 0 will be at p1 + w1 × p2 / (w2−w1). If w really was linear over this range then this would be an exact solution for w = 0, which, from Fermat's theorum we know is impossible.
If a / b is a good approximation for w1 / (w2−w1) then (bx, by, bz) could be a good candidate for a low integer value of w. It seems that to a first approximation we need
|a / b − w1 × p2 / (w2−w1)| < 1 / b3.
For b = 2 we get the range 1/2 ± 1/16 (0.4375 → 0.5625). For b = 3 there are two intervals: 0.3148 → 0.3519 and 0.6481 → 0.6852). The sums for each b value are (b−1)/b3 so the total is
total = Σ(b−1)/b3 = Σ1/b2 − Σ1/b3 = π2/6 − 1.2021… ≅ 0.4428
about 0.4428. So naively, if we have two good values we have a 44% chance of being able to find a better one. You can use the controls below to generate a table showing all the intervals up to a given depth. Click on "Full table".
Unfortunately a lot of these ranges overlap, for instance the range generated by 2/4 is entirely inside the range generated by 1/2. If we run the previous calculation removing fractions a/b that can be reduced we get a much smaller set. Click on the "Reduced table" to see the result. After 64 levels the reduced total is running about 0.07 behind the full value. It is never going to make up this defecit so the final total will probably be around 37%.
In addition there is the possibility that some of the intervals for different 'b' values will overlap, you can see this in the diagram below. Choose a depth setting and click on the "Overlaps" button to show where the ranges rall fall within the interval 0-1. The orange bars are ranges where a and b in a/b have a factor in common. They will fall entirely within the range of the reduced fraction which will appear in yellow higher up the diagram.
You will also be able to see that some of the yellow bars also overlap the ranges defined by bars higher up the chart, so the total coverage will be less than the 37% we estimated earlier.
(c) John Whitehouse 2019