Bilateral filters and arithmetic discrepancy in the original Collatz permutation

We study the positive integers that do not descend below their initial value when the permutation of the positive integers considered by Lothar Collatz in 1932 is iterated in either direction. The integers satisfying this condition at every depth are exactly the minima of the orbits of the permutation, so that each orbit contributes exactly one of them. This set was considered by M. Tavares (Discrete Mathematics, 2013). The paper studies its finite-depth approximations. The affine inequalities are replaced by conditions on the principal multipliers of the branches traversed, and it is proved that, at fixed depth, this replacement changes only a finite set. Each forward branch word corresponds to exactly one residue class modulo a power of 3, and each backward word to one class modulo a power of 2. By the Chinese remainder theorem, the density of each filter is the product of two survival probabilities of random walks with positive drift. A duality between the two walks provides exponential martingales, from which monotone rational bounds enclosing the limit of the densities are obtained. This limit is at least 1/36, and the bounds at depth 1000, computed in exact arithmetic, place it between 0.049779559058 and 0.049779559193. Consequently, the number of orbits whose minimum does not exceed a given value is asymptotically at most proportional to that value. It is also proved that the convergence is exponential, with a ratio common to both walks. The second part deals with the periodic discrepancy of the filters. It comprises a spectral identity for its quadratic energy, the expression of this energy in terms of correlations that factorize by the Chinese remainder theorem, and general bounds. Comparing each filter with one of smaller depth gives a bound for the discrepancy that is sublinear with respect to the period, although far from the exact values, which are given up to depth 7. In the forward direction, the passage from one depth to the next is described by a set of rejected residues. These residues admit an explicit parametrization, from which follow an almost antipodal pairing, with error at most one on all semicircles, and a recurrence for their exponential sums. Their relative equidistribution is equivalent to the decay of these sums at each fixed frequency, which is proved for part of the frequencies. An exact example of depth three and tables of the rejected sets up to step 20 are included. The results concern finite depths and their limit. The order of growth of the discrepancy, the relative equidistribution of the rejected residues, and the existence of the density of the set of orbit minima, together with its possible equality with the limit of the filters, remain open. The computational supplement contains the Python programs and exact tables supporting the numerical values. This is the English version of the Spanish original.