Math RefereeA proof of the positive density conjecture for integer Apollonian circle packings
Jean Bourgain and Elena Fuchs · Human-authored comparison
The proposed argument
The manuscript expresses curvatures in tangent-circle subpackings using shifted binary quadratic forms. It selects a logarithmic-scale family of shifts, lower-bounds the total number of represented integers, and upper-bounds overlaps using a quaternary-form count, local densities and congruence estimates. Combining single-set mass with the overlap bound yields a positive-density lower bound.
Human-authored comparison. The primary input is the 20-page arXiv v1; a distinct 25-page author-hosted copy is preserved separately. The journal text was not inspected. Source-reference edges are proposed dependencies, not checked external theorems.
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Your independent assessment
Check the packing hypotheses, shifted-form parametrization, shift-family selection, representation counts, overlap bounds and final inclusion–exclusion. Track interval lengths, residue hypotheses and external inputs.