In this puzzle, each square must contain a positive integer. All rows, columns and boxes must contain an arithmetic progression of length 9.
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Is it the following?
Very broadly, other than standard sudoku solving techniques:
for each row,col,square find the largest common divisor of all the numbers known so far, the step size has to divide that number.
in some cases also considering the spread, if, as for example in column 4, there are two numbers more than eight apart, the step size must be larger than one. As the lcd is 2 there, we know this column must hold 1,3,5,...,17.
We can use this kind of info in much the same way as with standard sudoku to rule out some possibilities, for example from what we know about column 4 we now can tell that the step in the bottom center square must be one, not seven. From this it follows that column 5 also cannot have step 7 but must have step 1.
intersection of ranges is another motive: now that we know bottom center square has range 1,..,9 and the middle column has either range (a) 7,...,15 or (b) 8,...16, we can tell that the intersection of these two shapes must contain 9,8, and since there are three slots 7, ruling out b.