Lower bounds for sets without perfect-power differences
Let D_k(N) be the largest size of a subset of {1, ..., N} with no positive k-th-power difference. We prove D_k(N) >= c_k N^alpha_k for every N >= 1, with positive constants c_k and exponents 0.75806770413, 0.9142, and 0.95295 for squares, fourth powers, and sixth powers. The proofs use a general interval-moment criterion for every positive integer k. We also prove arithmetic lifting results, exact product and prime-depth laws for interval capacity, and existence of the binary square capacity limit. The square construction refines Naslund's published example while retaining its binary…
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