Constants used in the n^{1/7}/18 bound

Replay: ./run_all.sh (exit 0).

Machine-checked

• Bedert Lemma 7.2, all 32 five-bit windows, exact in Q(sqrt(2)): verify_lemma72.py. Isolation gap min_{B_t}(-Im rho) - max Re rho = sqrt(2)/2.
• Rational enclosure 7/5 < sqrt(2) < 99/70 (because 49 < 50 and 9801 > 9800).
• Quadratic z^2 <= A z + B with A = 140 + 224 sqrt(2), B = 980 + 1856 sqrt(2), then Cstar <= 215816 and 2048 * 215816 = 441991168 <= 18^7 = 612220032. Scripts: track_constants.py, verify_certificate.py.

Human-checked inequality chain (arithmetic packed in lemma71_bounds.py)

Notation as in Bedert §3–§7. Symmetric A = -A subset Z\{0}, |A| = N, hat{1}_A + K >= 0. Auxiliary:

• f_t = 2 (1 + sin(2 pi t x)) hat{1}_A, so ||f_t||_min <= 4K.
• f_t * f_t >= -16 K^2.
• r_t = (1 - cos(2 pi t x + pi/4)) (f_t * f_t), so r_t >= -32 K^2.
• phi_t, psi_t even/odd parts: |phi| <= psi + 32 K^2.

Lemma 7.1 split of hat{1}_{B_t} = Q1 + Q2, for K >= 0:

• hat{1}_A = T1 + T2 with T1 = max(hat{1}_A, 0), T2 = min(hat{1}_A, 0), ||T1||_1 <= K, ||T2||_inf <= K, ||T2||_1 <= K, |T1| <= hat{1}_A + K.
• Convolution identity (hat{1}_A + K)*(hat{1}_A + K) = hat{1}_A + K^2.
• Young: ||u * v||_inf <= ||u||_1 ||v||_inf.
• Overflow split: if |S| <= hat{1}_A + K^2, write S = U + V with |U| <= hat{1}_A + K and ||V||_inf <= max(K^2 - K, 0) <= K^2.
• Collecting the even piece: |R1| <= 4(hat{1}_A + K), ||R2||_2 <= 20 K^2 - 6K.
• Odd piece: |S1| <= 2(hat{1}_A + K), ||S2||_2 <= 8 K^2 - 2K.
• Q1 = R1/2 + S1/2, so |Q1| <= 3(hat{1}_A + K), ||Q1||_1 <= 3K.
• Q2 = R2/2 + S2/2, so ||Q2||_2 <= 14 K^2 - 4K <= 14 K^2.
• Hermitian average Q_j(x) |-> (Q_j(x) + conj(Q_j(-x)))/2 preserves both bounds because hat{1}_A is even.

Proposition 7.3 test against H * H, H = |Q1|, then produces |B_t| <= Cstar K^4 with Cstar <= 215816.

Roth + AP:

• If N < 2 K^2 then K > sqrt(N/2).
• If N >= 2 K^2 some t in A has |A_t| >= N/(2K) (Roth, Lemma 4.1).
• Longest AP in A has length L <= 8 K^2 + 8 (Ruzsa / Bedert Lemma 4.2–4.4 for L >= 9; for L <= 8 the bound is immediate).
• Lemma 5.3: |B_t| >= |A_t| / L >= N / (16 K (K^2 + 1)).

Packing: N <= 16 Cstar (K^7 + K^5). Hence

• if K >= 1, N <= 32 Cstar K^7;
• if K < 1, N <= 32 Cstar K^5, so N / (32 Cstar) < 1 and K >= (N/(32 Cstar))^{1/5} >= (N/(32 Cstar))^{1/7}.

In all energy-branch cases K >= (N / (32 Cstar))^{1/7}.

Positive-integer form: A = B union -B, |B| = n, N = 2n, hat{1}_A(x) = 2 sum_{b in B} cos(2 pi b x), so Chowla K(n) >= K_sym(2n)/2. The energy-failure branch is K(n) >= sqrt(n)/2. Both branches are at least n^{1/7}/18 because 2048 * 215816 <= 18^7 and sqrt(n)/2 >= n^{1/7}/18 for every n >= 1.