Square-difference-free sets of exponent 0.7580758
Let $D(N)$ be the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no two elements whose difference is a nonzero perfect square. We give a computer-assisted construction proving $D(N)\ge N^{0.7580758318008816-o(1)}$, improving the exponent $0.752796455874514\ldots$ of Krachun's ranked-block construction. The residues we use carry subintervals of $[0,1]$ that are ordered along every modular square difference. A finite stopping-word argument shows that if finitely many such alphabets, in pairwise coprime perfect-square moduli, have interval moments whose contributions sum to more…
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