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We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee-Huang-Yang formula. This result was first established in 20 by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported, gives better rates and, in our opinion, is substantially simpler.
Basti et al. (2021) studied this question.