Carries and Cocycles

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Adds digits produce carries with probability 1/2 − 1/(2b) (45% when b = 10); choosing balanced coset representatives cuts forced carries to (b^2 − 1)/4 (≈ b^2/4) and that lower bound is sharp. For a normal subgroup with coset set X, if the proportion of non‑carries exceeds 7/9 then X is a subgroup (the extension splits); this uses approximate homomorphism results (Coppersmith–Luby–Rubinfeld) and arithmetic‑combinatorics tools (Cauchy‑Davenport/Freiman). Open problems include exact maxima for c(X) (conjectured 9/16 for Z_b in Z_{b^2}).

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" chapters (10)

1. 0:00Introduction to carries
2. 2:22Carries and co-cycles
3. 5:30Balanced representatives
4. 9:30Bounding carries
5. 12:32Group extensions and splitting
6. 17:02Approximate homomorphisms
7. 22:22Proving the seven-ninths theorem
8. 30:23Arithmetic combinatorics
9. 36:48Counterexamples and problems
10. 47:35Discussion and Q&A