| from mpmath import mp, mpf, matrix, power, eye, det, nstr |
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| |
| |
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|
| def _build_matrix(N, s, M): |
| """ |
| M×M monomial-basis truncation of the Ruelle transfer operator L_N^(s). |
| |
| [L_N^(s) x^j](x) = sum_{n=1}^{N} (n+x)^{-(2s+j)} |
| |
| Expanding (n+x)^{-alpha} = sum_{i>=0} (-1)^i * (alpha)_i/i! * n^{-(alpha+i)} * x^i: |
| |
| A[i,j] = (-1)^i * (2s+j)_i / i! * sigma_{j+i}(s) |
| |
| where sigma_k(s) = sum_{n=1}^{N} n^{-(2s+k)}. |
| |
| d(N) is the zero of det(I - A_M(s)), the Fredholm determinant approximation. |
| """ |
| sigma = [] |
| for k in range(2 * M): |
| alpha = 2 * s + k |
| sigma.append(sum(power(mpf(n), -alpha) for n in range(1, N + 1))) |
|
|
| A = matrix(M, M) |
| for j in range(M): |
| alpha_j = 2 * s + j |
| poch = mpf(1) |
| fact = mpf(1) |
| for i in range(M): |
| if i > 0: |
| poch *= (alpha_j + i - 1) |
| fact *= i |
| coeff = poch / fact |
| if i % 2 == 1: |
| coeff = -coeff |
| A[i, j] = coeff * sigma[j + i] |
| return A |
|
|
|
|
| def compute(N, M=70, dps=25): |
| """ |
| Compute d(N) = Hausdorff dimension of |
| E_N = {x in [0,1] : all continued-fraction partial quotients of x are <= N} |
| |
| Method: bisect on the sign of det(I - A_M(s)), the Fredholm determinant |
| approximation. At s < d(N) the sign is -1; at s > d(N) it is +1. |
| Accuracy is ~(M/3) significant digits for N=2; M=70 gives ~24 digits. |
| |
| Parameters |
| ---------- |
| N : int, N >= 2 |
| M : matrix truncation size (default 70 for ~24 digits) |
| dps : working decimal precision (should exceed M/3) |
| |
| Returns |
| ------- |
| mpf : d(N) to approximately min(dps, M/3) significant digits |
| """ |
| mp.dps = max(dps, M // 2) + 20 |
|
|
| s0_map = {2: "0.531", 3: "0.731", 4: "0.819", 5: "0.870"} |
| s0 = mpf(s0_map.get(N, str(round(1.0 - 6.0 / (3.14159265 ** 2 * N), 3)))) |
| s_lo = s0 - mpf("0.1") |
| s_hi = s0 + mpf("0.1") |
|
|
| sign_lo = 1 if det(eye(M) - _build_matrix(N, s_lo, M)) > 0 else -1 |
|
|
| tol = mpf(10) ** (-(dps + 5)) |
| while s_hi - s_lo > tol: |
| s_mid = (s_lo + s_hi) / 2 |
| d = det(eye(M) - _build_matrix(N, s_mid, M)) |
| if (1 if d > 0 else -1) == sign_lo: |
| s_lo = s_mid |
| else: |
| s_hi = s_mid |
|
|
| return (s_lo + s_hi) / 2 |
|
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|
|
| if __name__ == "__main__": |
| for N in [2, 3, 4, 5]: |
| val = compute(N, M=70, dps=25) |
| print(f"N={N}: {nstr(val, 25)}") |
|
|
