*[This is notation (common in model theory) for the union of $B$ and $C$. -T]*

@Dear Prof Tao , Suppose we have a point p(a,b) a and b are given and let L be a line s.t p(a,b) belong to L

By this information can we know the equation of L in terms of P(a.b) ?can it gives as more information about L?

Ah, this claim turns out to be not quite true as stated; one needs to use a non-isotropic dilation in which the x and y coordinates are multiplied by different dilation factors, rather than the isotropic dilation currently written, in order to have a positive probability of the rectangle being approximately dyadic. Basically a horizontal dilation will make (with positive probability) the horizontal side length close to a power of two (say between $1.001$ and $1.002$ times a power of two), a vertical dilation will achieve a similar effect for the vertical side length, and then a random translation will then place the rectangle very close to a dyadic rectangle with positive probability.

]]>Ah, right, this condition is in fact never used, it can be safely deleted.

]]>Thank you.

]]>It means that the incidence relation can be defined using a finite number of linear inequalities (over some suitable ordered ring). (Here the “semi-” prefix is in the same sense as “semi-algebraic set“.) For instance, to describe the incidences between a set of points in the plane and a set of rectangles, one can view as a collection of pairs in , and as a collection of quadruples in , and the incidence relation is then given by the linear inequalities and .

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