Math RefereeOn the Local-Global Conjecture for Integral Apollonian Gaskets
Jean Bourgain and Alex Kontorovich · Appendix by Péter P. Varjú
The proposed argument
The manuscript encodes circle curvatures in an Apollonian-group orbit and constructs a weighted representation function from shifted binary quadratic forms. It reduces a power-saving exceptional-set bound to a major-arc lower bound and a minor-arc mean-square estimate. Uniform spectral gap and effective bisector counting supply the major-arc equidistribution; explicit exponential-sum estimates handle three denominator ranges on the minor arcs. A separate appendix derives the needed uniform spectral gap from arithmetic subgroups. The final comparison of represented mass and exceptional-set size gives density one among admissible integers.
Frozen arXiv 1205.4416v2 (14 May 2013), including the appendix. The 2014 journal typesetting was not inspected. External proof inputs are constrained to the recorded received date, 24 May 2012. Reading materials and panel findings are separate.
Reading order
- 01
Arithmetic encoding and congruences
Descartes quadruples, group parametrization and admissible residue classes.
- 02
Reduction to circle-method estimates
Representation counts, truncation and exceptional-set comparison.
- 03
Major arcs
Uniform spectral counting, singular series and truncation transfer.
- 04
Minor arcs
Control three denominator ranges and combine mean-square estimates.
- 05
Spectral-gap appendix
A bounded-generation and subgroup-gap argument for the Apollonian congruence family.
Follow the dependencies
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Your independent assessment
Trace the representation-function reduction through the spectral and congruence inputs, major arcs and three minor-arc ranges. Distinguish vector from projective stabilizers, check uniformity and scale choices, and record exactly which external hypotheses each step requires.