From https://mathenchant.wordpress.com/2020/08/16/the-muffin-curse/ (slightly adapted):
Imagine an infinite supply of 1-ounce chocolates rolling off an assembly line that's staffed by two immortal and indefatigable employees (call them Ethel and Lucy) who have to feed an infinite line of students. When a new chocolate arrives, Ethel cuts it up and puts pieces into an infinitely large holding area; meanwhile, Lucy takes pieces from the holding area and hands them out to students. All the chocolate pieces must be handed out to students, and each student must get exactly *x* ounces of chocolate. If Lucy and Ethel want the smallest piece any student gets to be as large as possible, what goal should they shoot for, as a function of *x*? I'm pretty sure that when *x* is rational, this is just the muffin problem discussed in this forum back in 2008. But what happens when *x* is irrational? If, say, *x* is the golden ratio ϕ = (1 + sqrt(5))/2 = 1.618..., is there a way for Ethel to divide the 1-ounce chocolates into smaller morsels, and for Lucy to distribute the morsels, so that every morsel gets eaten, and every student gets exactly ϕ ounces of chocolate, and no morsel is less than 0.4 ounces? It may be relevant to mention that for the original muffin problem, f(m,s) (defined as the maximum possible size of the smallest piece among all ways of distributing m muffins among s students) has the property that f(m,s) = g(m/s) for a certain function g that is NOT continuous as a function from the rationals to the rationals (here I am using the epsilon-delta definition of continuity, which makes sense even when the domain of a function is the rationals). So g(.) does not admit an extension to an everywhere-continuous function from the reals to itself (in case you were hoping, as I was, that this would be the answer). There may be some sort of semicontinuous extension at work here. I chose 0.4 because if we let s and m be consecutive Fibonacci numbers, then g(m/s) seems to be converging to something slightly larger than 0.4. Here's Richard Chatwin's article: Richard Chatwin, An optimal solution for the muffin problem, https://arxiv.org/pdf/1907.08726.pdf. Jim Propp