An Exponent of 1.043 for the Unit Distance Problem
Let $u(U)$ count the unordered pairs at distance one in a finite planar set $U$. We construct finite sets $U_j$ with $|U_j|\to\infty$ and $u(U_j)/|U_j|^{1.043171}\to\infty$. The largest exponent previously claimed, apart from preliminary versions of the present work, is $1.0358$, in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin. Here unit distances come from elements of relative norm one in quadratic extensions $K/F$ in which $F$ has both real and complex places, and the fields come from an infinite pro-$2$ tower over $\Q(\sqrt{241})$,…
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