holds with $\rho+\rho^2$ replaced by $\rho^{4/3}+\rho^{5/3}$ which have the scaling of the Coulomb and kinetic energies, respectively. The higher power $\rho^2$ arises from the trial state used in our upper bound (Proposition \[prop:upper\_bound\]) and it is used to control some errors appearing when merging quantum systems with overlapping supports. This is explained in Section \[sec:upper\_bound\] below. Finally, the last gradient term in  is used to control local variations of $\rho$ in $L^{\infty}$. One could expect gradient errors involving only $\int_{{\mathbb{R}}^3}|\nabla\sqrt\rho|^2$ and $\int_{{\mathbb{R}}^3}|\nabla\rho^{1/3}|^2$ which are believed to arise in the gradient expansion of the uniform electron gas for, respectively, the Coulomb and kinetic energies.

One interesting case is when the density is given by a fixed function $\rho$ with $\int_{{\mathbb{R}}^3}\rho=1$, which is rescaled in the manner $$\rho_N(x)=\rho(N^{-1/3}x).$$ After taking ${\varepsilon}=N^{-1/12}$ in  we obtain the following simple bound $$\begin{gathered}
 \left|E(\rho_N)- N\int_{{\mathbb{R}}^3}e_{\rm UEG}\big(\rho(x)\big)\,dx\right|\\
 {\leqslant}CN^{\frac{5}{12}}\int_{{\mathbb{R}}^3}|\nabla\sqrt\rho(x)|^2\,dx
+C N^{\frac{11}{12}}\int_{{\mathbb{R}}^3}\left(\rho(x)+\rho(x)^2+|\nabla\rho^\theta(x)|^p\right)\,dx.
 \label{eq:LDA_main_rescaled_density}\end{gathered}$$ It is conjectured [@Kirzhnits-57; @LanPer-80; @LanMeh-83; @LevPer-93; @EngDre-11] that the next order in the expansion of $E(\rho_N)$ should involve the gradient correction to the kinetic energy $$\int_{{\mathbb{R}}^3}\left|\nabla\sqrt{\rho_N}(x)\right|^2\,dx=N^{\frac13}\int_{{\mathbb{R}}^3}\left|\nabla\sqrt\rho(x)\right|^2\,dx$$ and the gradient correction to the Coulomb energy $$\int_{{\mathbb{R}}^3}\left|\nabla\rho_N^{1/3}(x)\right|^2\,dx=N^{\frac13}\int_{{\mathbb{R}}^3}\left|\nabla\rho^{1/3}(x)\right|^2\,dx.$$ In particular, the next order should be proportional to $N^{1/3}$. It remains an open problem to establish this rigorously.

In 