The Misleading Count

The Markoff spectrum measures how well indefinite ternary quadratic forms approximate zero. For a form F with discriminant D, define μ(F) as the cube of the smallest nonzero value |F(x)| over integer vectors, normalized by |D|. Forms with large μ(F) — those that stay far from zero — are arithmetically rigid. The Markoff spectrum is the set of these values. Margulis asked: how many spectral points μ ≥ 1/X appear as X grows?

Martini’s numerical thesis computed the count up to moderate X and found growth that looked quadratic — approximately 1.2X². The data fit the curve. The exponent appeared to be two.

Gamburd, Ghosh, Sarnak, and Whang prove the count grows as γX log X, where γ is an explicit positive constant given by a convergent series. The exponent is one, not two. The logarithmic factor means the function grows faster than linear but slower than any power above one. At the scales Martini’s computation reached, X log X and X² are visually indistinguishable — both curve upward, both accelerate. But their asymptotics diverge completely. By X = 10⁶, the ratio X²/(X log X) exceeds 70,000. The computation was in a pre-asymptotic regime where the true growth rate had not yet asserted itself.

The proof resolves a companion problem of Serre’s on isotropic ternary forms — the density of forms admitting a nontrivial integer zero — with an explicit asymptotic involving X⁶/√(log X) and p-adic probability constants. Both problems required developing techniques for high ramification, where the discriminant factors into heavily divisible parts whose local structure resists standard sieve methods.

The numerical evidence was honest. The data genuinely curved upward. The conjecture was reasonable. But the exponent was wrong by one, and the logarithm was invisible at computable scales. The convergence to the true asymptotic was slower than the computation.