Mathematics · Number theory and dynamics Collatz research literature
Tao's almost-all Collatz theorem uses a decay estimate $|\chi_n(\xi)|\ll_A n^{-A}$ for the characteristic function $\chi_n(\xi)=\mathbb{E}\,e(\xi\,\mathrm{Syrac
Tao proved that almost all Collatz orbits attain almost bounded values in the sense of logarithmic density, and asked whether natural density is attainable. All
For the odd Collatz map, we prove that every positive odd unit $a\le M$ receives at least $cX/M$ sources from $(X/4,X]$ whose first landing at or below $M$ is $
For the coupled real and ternary Syracuse affine series, we prove a Gaussian total-variation transition modulo $3^k$ under the real-tail event $R>r$. Classical
The fine ternary digits of a Syracuse random affine offset remain superpolynomially mixed after retaining the exact total division exponent as a second coordina
Let $\pi_a(x)$ be the number of $n\le x$ whose Collatz orbit contains $a$. Wirsching (2003) asked whether $\pi_a(x)\ge c\,x/a$ holds with one absolute constant
For the accelerated Collatz map $U$, we study certificates in which a smaller positive odd start reaches an observed forward endpoint. If a forward word uses on
We establish five unconditional results on the Diophantine, $(2,3)$-adic, and block-renormalization structure of hypothetical positive Collatz cycles. First, sy
We prove effective restrictions on the periodic-pattern complexity of every positive Collatz trajectory. Fix a finite dictionary of exponent words. Count each f
We extend published Collatz synchronization results with quantitative estimates for conditioned clocks, couplings that retain actual inputs, and lower bounds fo
For the $n$-step Syracuse random variable $\mathrm{Syrac}_n = \sum_{m=1}^n 3^{m-1} 2^{-s_m} \pmod{3^n}$ with characteristic function $\chi_n(\xi)$ and primitive
Tao proved that the odd $N$ with $\mathrm{Syr}_{\min}(N)>N_0$ have logarithmic density $O(\log^{-c}N_0)$ for a small absolute $c>0$, and earlier work on this si
We study smaller positive sources that share a Collatz future with an original source before any forward iterate of the latter falls below it. Every finite pari
We study the stationary law of the fair Collatz affine maps acting jointly on the real and ternary coordinates. Leen's counted-coalescence estimate yields sharp
Krasikov and Lagarias (2003) proved that feasibility of a linear program $L_k^{NT}(\lambda)$ implies $\pi_a(x)\gg x^{\log_2\lambda}$. Whether $\lambda_k\to2$, i
We prove the non-existence of non-trivial positive integer Collatz cycles with $m = 94$ or $m = 95$ local-minimum blocks ($m \le 95$), conditional on Barina's $
We study distinct Syracuse orbit points in finitely many fixed logarithmic bands before first entry below $A^{.99}$. Taking the source natural-density limit fir
We extend restrictions on positive rational shortcut-Collatz orbits from automatic parity and two primitive-substitution classes to every coded primitive fixed
We study positive integer cycles of the accelerated Syracuse map $U(x)=(3x+1)/2^{v_2(3x+1)}$, focusing primarily on the restricted binary valuation alphabet $\m
Any $k$-step cycle of the Syracuse Collatz map with valuation vector $a = (a_0, \dots, a_{k-1})$ and division count $S = a_0 + \dots + a_{k-1}$ satisfies $(2^S
Can unbounded prime-factor counts repair pointwise Collatz descent certificates based on size and finite modular information? We prove that, for every modulus $
