A New Bound for the Gauss Circle Problem

Abstract: Let $E(X)=#{(a,b)\in\Z^{2:a^{2+b^2\le}} X}-πX$. We prove that $E(X)=O_\varepsilon(X<sup>{209/670+\varepsilon})$ for every $\varepsilon&gt;0$, uniformly for real $X\ge4$. This breaks the $5/16$ threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements: a cone distribution estimate that preserves separate fourth- and sixth-power contributions in the double large sieve, and an explicit inversion of Taylor-coefficient constraints that cancels the linear error in a pair-counting argument. These refinements, together with the Robert--Sargos and Watt spacing estimates and the Vinogradov mean value theorem, yield the improvement.