Almost all Collatz orbits attain almost bounded values

This post is messing with my self image and the false front I show the world. Your words challenge my image and that is more than uncomfortable. I will tell you what’s comfortable for me. I am comfortable thinking of myself as intellectually superior to the hoi polloi, the kind of person who snootily uses terms like “hoi polloi” in her writing. I don’t tell anyone I have to look up the words to make sure I am using them correctly bc I am not actually the walking dictionary I pretend I am–but you did’t know my truth. In fact, I did’t even know my truth until I (attempt) to read your work. My truth is that I have potential sitting within my mind and soul that is untapped bc I have been doing things other than improving my education and intellect. I have a long way to go before I begin to resemble the person I want you to think I am. I read (attempted to read) your writing and fell into self-condemnation as naturally as I breathe. But then, I remembered, thoughts don’t just happen to me, I a choosing them, and I don’t have to punish myself. I can soothe my hurt ego knowing that I have a chance to be more, to do more–to live my dream of developing myself intellectually and even more importantly, spiritually. As long as I am still breathing I can start down my desired path by reading (attempting to read) the words you have so generously shared. I can follow your links to worlds of ideas heretofore unknown to me, and revel in the fact that I am still alive and I still have a chance. A chance to be more like your professional self and less like the person depending on an ivy league college degree obtained a quarter of a century ago, to validate my worth as a person. I cringe as I share with that level of honesty because I do not want to be that person–not when others can see. Yes, I only just realized that I was the last to know that my haughty front was just a facade and not supported by recent efforts. You have read my confession, seen me acknowledge the wasted mind that is my shame and now you get the privilege of seeing how your incomprehensible math/science writing has motivated me to change. I have heard that if you are the smartest person in the room, you are in the wrong room. I have lingered in the wrong room but now I exit in search of an environment where I am as clueless as I was when I read (attempted to read) your pst because I know I can get there from here. Thanks for you post. It woke me up. I’ll let you know when I have advanced enough to read it with understanding. Lol! Sincerely, Caroleena, the expert escort

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I’ve just uploaded to the arXiv my paper “Almost all Collatz orbits attain almost bounded values“, submitted to the proceedings of the Forum of Mathematics, Pi. In this paper I returned to the topic of the notorious Collatz conjecture (also known as the $latex {3x+1}&fg=000000$ conjecture), which I previously discussed in this blog post. This conjecture can be phrased as follows. Let $latex {{bf N}+1 = {1,2,dots}}&fg=000000$ denote the positive integers (with $latex {{bf N} ={0,1,2,dots}}&fg=000000$ the natural numbers), and let $latex {mathrm{Col}: {bf N}+1 rightarrow {bf N}+1}&fg=000000$ be the map defined by setting $latex {mathrm{Col}(N)}&fg=000000$ equal to $latex {3N+1}&fg=000000$ when $latex {N}&fg=000000$ is odd and $latex {N/2}&fg=000000$ when $latex {N}&fg=000000$ is even. Let $latex {mathrm{Col}_{min}(N) := inf_{n in {bf N}} mathrm{Col}^n(N)}&fg=000000$ be the minimal element of the Collatz orbit $latex {N, mathrm{Col}(N), mathrm{Col}^2(N),dots}&fg=000000$. Then we have

Conjecture 1 (Collatz conjecture) One has $latex {mathrm{Col}_{min}(N)=1}&fg=000000$ for…

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