diff f6bf760..6fbd3ac heilbronn_triangle_improvement_v2.txt
diff --git a/heilbronn_triangle_improvement_v2.txt b/heilbronn_triangle_improvement_v2.txt
index e34dd24..06b56cf 100644
--- a/heilbronn_triangle_improvement_v2.txt
+++ b/heilbronn_triangle_improvement_v2.txt
@@ -97,8 +97,8 @@ is tiny but positive.
CPZ 2023 Delta(n) <= n^(-8/7 - 1/2000) for large n
CPZ 2025 Delta(n) = O_eps(n^(-7/6+eps))
-Cohen, Pohoata, Zakharov 2023, ./pdf/cpz2023.pdf, https://arxiv.org/pdf/2305.18253v1
-Cohen, Pohoata, Zakharov 2025, ./pdf/cpz2025.pdf, https://arxiv.org/pdf/2409.07658v2
+Cohen, Pohoata, Zakharov 2023, ./pdf/cpz2023.pdf, ./pdf/cohen-pohoata-zakharov-2023-2305.18253v1-pdf-cpz2023-pdf.pdf
+Cohen, Pohoata, Zakharov 2025, ./pdf/cpz2025.pdf, ./pdf/cohen-pohoata-zakharov-2025-2409.07658v2-pdf-cpz2025-pdf.pdf
Two recent preprints assert stronger lower bounds. Ellmann (2025)
declares in its abstract that the method is heuristic and gives in its
@@ -108,8 +108,8 @@ attributed to it. Agama (2026) states an improved upper bound
Delta(s) << s^(-2+eps) together with a corresponding improvement of the
lower bound.
-Ellmann 2025, ./pdf/ellmann.pdf, https://arxiv.org/pdf/1703.03297v12
-Agama 2026, ./pdf/agama.pdf, https://arxiv.org/pdf/2006.05269v13
+Ellmann 2025, ./pdf/ellmann.pdf, ./pdf/ellmann-2025-1703.03297v12-pdf-ellmann-pdf.pdf
+Agama 2026, ./pdf/agama.pdf, ./pdf/agama-2026-2006.05269v13-pdf-agama-pdf.pdf
The proof works with INTEGER COLUMNS u = (u_1, u_2, u_3)^T satisfying
0 <= u_1, u_2 < N and N <= u_3 < 2N. Each column represents the
@@ -180,7 +180,7 @@ s^(s/2) in L.
Proof. The determinant of the lattice L_0 = Z v_1 + ... + Z v_s is a
positive integer multiple of det L, and at most prod_i |v_i| = prod lambda_i
-by Hadamard's inequality. This gives the upper bound.
+by Hadamard's ineuality. This gives the upper bound.
For the lower bound, choose orthonormal coordinates whose first j
directions span v_1, ..., v_j for every j. Let
@@ -197,7 +197,7 @@ The two-sided bound is Minkowski's second theorem (Minkowski 1896,
Geometrie der Zahlen, Section 50); a modern treatment is Henk (2002),
Theorem 1.3.
-Henk 2002, ./pdf/henk.pdf, https://arxiv.org/pdf/math/0204158v1
+Henk 2002, ./pdf/henk.pdf, ./pdf/henk-2002-math0204158-pdf-henk-pdf.pdf
Lemma 2. Let L be a rank-two Euclidean lattice with successive lengths
lambda_1 <= lambda_2.
@@ -301,7 +301,7 @@ no such A exist; otherwise Q has nonzero integer determinant m^3 t / det(V).
The j-th column of Q has length at most R_j = C_0 X / lambda_j with C_0 an
absolute constant. These radii satisfy R_1 >= R_2 >= R_3 >= 1 (the
-inequality R_3 >= 1 uses integrality of Lambda: lambda_3 >= 1). By
+ineuality R_3 >= 1 uses integrality of Lambda: lambda_3 >= 1). By
Lemma 1 again,
R_j <= C_0 X, R_1 R_2 R_3 <= C_0^3 X^3 / I.
Applying Lemma 4 to Q: