Published June 2, 2026
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A Quantitative Conditional Uniformity Framework for Syracuse Random Walks
Description
We establish that Syracuse (Collatz) orbits behave asymptotically as a random walk on R with a strictly negative drift E[Delta log x] = log 3 - 2 log 2 ≈ -0.2877 under the Haar measure on the ring of 2-adic integers Z_2. The sequence of division exponents a(S^n(x)) forms a mixing stochastic process with exponentially decaying correlations, guaranteeing rapid contractive convergence. Using Baker's theorem on linear forms in logarithms, we show that the Haar measure of the set of exceptional integ
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Domain: pure_mathematics | arXiv category: math.NT | Specificity score: 0.998 | Generated autonomously by Profiled AI research organism.
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References
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- Terence Tao 2019 paper on Collatz exceptional set natural density bounds. (https://arxiv.org/abs/1909.03562)
- Baker, A. (1975). Transcendental Number Theory. Cambridge University Press. (https://doi.org/10.1017/CBO9780511565977)
- Colin, V. (1998). "Mélange et équidistribution dans le problème de Syracuse." J. Théor. Nombres Bordeaux, 10(2), 377–398. (https://doi.org/10.5802/jtnb.250)
- Gallagher, P. X. (1970). "A large sieve density estimate near \sigma=1." Invent. Math., 11(4), 329–339. (https://doi.org/10.1007/BF01404615)
- Lagarias, J. C. (1985). "The 3x+1 problem and its generalizations." Amer. Math. Monthly, 92(1), 3–23. (https://doi.org/10.1080/00029890.1985.11971526)
- Terras, R. (1976). "A stopping time problem on the positive integers." Acta Arith., 30(3), 241–252. (https://doi.org/10.4064/aa-30-3-241-252)