A New Bound for the Gauss Circle Problem

Let $E(X)=\mathrm{\#}\{(a,b)\in {\mathbb{Z}}^2:a^2+b^2\le X\}-\pi X$. We prove that $E(X)=O_{\epsilon }(X^{209\mathrm{/}670+\epsilon })$ for every $\epsilon >0$, uniformly for real $X\ge 4$. This breaks the $5\mathrm{/}16$ 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.