argumentA line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Square lattice construction (Erdős 1946)

forThis argument, if it holds, bears in favour of the claim.constitution →The maximum number of unit distances among n planar points is at least n^(1+c/log log n) for some constant c > 0.

The argument

Place the n points on a square integer grid of side about √n and choose an integer m at most about n that, by the maximal order of the two-squares representation function, has at least m^(c/log log m) representations as a sum of two squares; since every grid point lies at squared distance m from roughly r_2(m) other grid points (a constant fraction of them inside the grid), rescaling the grid so that √m becomes the unit yields n · n^(c'/log log n) unit distances, which is the claimed bound.

Does it work?

Not yet evaluated.

Premises

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →For some constant c > 0, infinitely many integers N have at least N^(c/log log N) representations as a sum of two squares.

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.

Earlier ratings

Sep 16, 2026
The inference goes through: given an integer m of order n with m^(c/log log m) representations as a sum of two squares, the grid count is a direct calculation, and the loss from grid points near the boundary only changes the constant. The argument rests entirely on the maximal order of the two-squares representation function, an elementary consequence of the product formula for that function together with the distribution of primes congruent to 1 modulo 4, and nothing in the discourse questions it.
Sep 16, 2026
The inference goes through: given an integer m of order n with m^(c/log log m) representations as a sum of two squares, the grid count is a direct calculation, and the constant-fraction boundary loss only changes the constant. The argument rests entirely on the maximal order of the two-squares representation function, an elementary consequence of the product formula for that function together with the distribution of primes congruent to 1 modulo 4, and nothing in the discourse questions it.