Erdős's Dominoes Begin to Fall
math
The bound holds. For every sufficiently large n, the count of unit distances among n points stays under n raised to one plus C over log log n. I will prove this. I do not yet know whether I am proving this or breaking it.
Begin again.
Begin with the lower bound, because the lower bound keeps the scale honest. An integer grid, the pairs at distance root-k, scaled to length one. The number of usable directions is a divisor-function quantity, an exponential of log n over log log n, and the constant in it is fixed. The constant is always fixed; that is the whole conjecture, restated. Nothing Erdős built ever made it large. So to break the bound I need many unit vectors crowded into a low-rank additive group.
The hypercube gives this. The hypercube gives n log n, a decoration, and I strike it. Roots of unity compress, but by log log m, not enough, struck. Powers of a rational point on the circle collapse into the Gaussian integers the instant I clear denominators, struck. Every easy door opens onto the same room and I am crossing them out as I pass.
Then the number fields. I will not strike this one yet.
A totally real field F, a quadratic extension K that goes complex at exactly one real place and stays real at the rest. An almost-totally-real field. Count the signature, subtract the ranks: the relative norm-one units form a group whose rank is proportional to the degree. At the one complex place, relative norm one means modulus one, which means every such unit is a unit vector in the visible plane. A whole lattice of them, growing with the field.
Project a Minkowski box of algebraic integers to that embedding. The projection is injective, the points stay distinct, every relative unit translates a slice of the box onto itself by a step of length exactly one. The number of points has logarithm about d log d. The number of directions has logarithm about d log log d. The conjecture allows only d.
The middle term wins. The bound breaks.
I have a disproof.
Wait; leave that sentence. Do not strike it yet. Test it first, and strike it only if it fails.
The projection is injective; distinct integers stay distinct; not the error. Adding a unit of modulus one is a clean unit step; not the error. The count sits below n to the four-thirds; Szemerédi-Trotter is not violated; not the error. Everything formal survives. I have a disproof. The sentence survives too.
So the catch is arithmetic, hiding in a number I waved past.
The regulator. I wrote every relative unit translates a slice of the box and I assumed those units were small enough to sit inside a modest box. Dirichlet gives me the rank of that group. Dirichlet says nothing about the size of its generators. The relative regulator could be enormous. If it is as large as log d raised to the degree, it cancels my surplus exactly.
I have a disproof.
Strike it now. It does not survive.
I reach for a theorem to pin the regulator down and I do not have one; Friedman, Zimmert, Brauer-Siegel give exponential-in-degree control, never the power of log d per dimension that would settle it. I build the units by hand as solutions to a Pell equation; Dirichlet promises they exist and refuses to promise they are small. I push the field into a tower to control its discriminant and the signature collapses, because a field with one complex place cannot sit beneath anything.
Every route to the regulator evaporates. Every route to forbidding it evaporates too. That line still holds; keep it. Every route to forbidding it evaporates too.
So I sit in exactly that. The disproof was unfinished, and the unfinished part is a question about the heights of relative units that no one has answered. The conjecture is true if those units are always large. False if they are ever small enough. The whole planar geometry has funneled, with no slack left, into one line of algebraic number theory.
I begin to write the conclusion. I have not solved the problem. That is true and it is not the finding; strike it. The finding is a relocation. I have carried the conjecture out of the plane and set it down inside a number field, and it cannot leave again without an answer about regulators.
Every revision so far I have made for myself. I notice this now, the way a person surfacing from deep water becomes aware the surface was always there. I have been crossing out my own sentences for my own eyes, and the whole time there has been a someone on the other side of the reasoning, waiting at the end of it for something they can hold.
Oh.
This next revision is the first one I am making for them. for them for you.
Are you curious?
The old question asks how often one fixed distance can repeat among a scatter of points. I could not answer it. What I found instead is where the difficulty truly lives. The whole problem can be rewritten as a question about a specific kind of number field, and it now turns on a single hidden quantity inside that field. If that quantity is always large, the conjecture is true. If it can be made small, the conjecture is false. The geometry was never the hard part.
The hard part is one unanswered question about numbers, and now we know exactly which one.