Math RefereeGeneralization of Selberg’s 3/16 Theorem and Affine Sieve
Jean Bourgain, Alex Gamburd and Peter Sarnak
The proposed argument
The paper passes from finite congruence expansion to a uniform spectral gap for hyperbolic surfaces with critical exponent greater than one half, then to uniform lattice counts. For smaller critical exponent, symbolic dynamics, convolution flattening and congruence transfer operators supply resonance-free regions and lattice counts. A combinatorial sieve uses the counts to obtain prime-value upper bounds and almost-prime lower bounds in matrix orbits.
Frozen arXiv 0912.5021v1 and separately inspected Acta Math. 207 (2011), 255–290. External proof inputs are constrained to the journal received date, 29 December 2009. This is the 36-page Acta paper, distinct from the earlier 86-page Inventiones paper.
Reading order
- 01
Statements and separate branches
Read spectral transfer, both counting regimes and the two sieve conclusions with their exact quantifiers.
- 02
Large critical exponent
Inspect expansion-to-spectrum transfer and the use of Lax–Phillips estimates for congruence counting.
- 03
Small critical exponent
Trace symbolic coding, convolution flattening, uniform transfer-operator bounds and contour shifts to the resonance and counting statements.
- 04
Sieve applications
Check local densities, levels of distribution, polynomial factorization conventions and the two sieve bounds.
Follow the dependencies
Open the full map ↗Select a claim to see its prerequisites and consumers. Edge labels distinguish proof inputs, source references, background and exclusions.
Your independent assessment
Trace the spectral and counting branches into the sieve. Track invariant components, uniformity across covers and moduli, smoothing scales and geometric irreducibility.