INTERSECTION OF A MORAN TYPE SIERPINSKI CARPET AND A LINE WITH RATIONAL SLOPE

The paper’s Theorem 2 states that for the Moran-type Sierpinski carpet Fσ generated by scales 3 and 4 under a control sequence σ, and for a line La with rational slope tan θ, the upper/lower box dimensions of Fσ ∩ La equal limsup/liminf growth rates of the row-sum of a product of (N+M)×(N+M) “transfer” matrices chosen by a greedy digit rule; this is set up via affine maps T^s_d(x)=m x + d1(M/N) − d2 on the intercept and a partition of Jθ into N+M intervals, then encoded by matrices A^j_s (with j the greedy digit) so that N_k(a)=∥e_{i0(a)}A^{ξ_1}_{σ_1}…A^{ξ_k}_{σ_k}∥, and finally dim_B is obtained by dividing by n0(k)log3+n1(k)log4 (statements and construction appear in Theorem 2 and the surrounding development). This matches the candidate’s solution, which gives a more explicit arithmetic “one-step transfer” lemma establishing the (p,q) transition rule q = 3p + j − (2M+2) + (d1 M − d2 N) for m = 3 and q = 4p + j − (3M+3) + (d1 M − d2 N) for m = 4, and then proves the counting/product formula and the comparability to covering numbers. Minor issues: the paper contains a likely typo in the A^j_1 definition (it writes q − 3p − j where 4p is required), and glosses over the covering-number comparability; the candidate corrects both by using the proper 4p and by supplying uniform comparability constants. Apart from notational slips (tan θ sometimes written M/N vs N/M) and those minor gaps, the approaches are essentially the same and yield the same formulas (see paper’s definitions of Fσ and Jθ, the matrix setup for A^j_s, the Γa partition, and the Nσ_k(a) to matrix-product identification). Therefore, both are correct with substantially the same proof, with the model filling in arithmetic details and boundary/tie-breaking conventions that the paper omits.