The first 128 digits of an autoconvolution inequality

Using rigorous high-precision floating point arithmetic we compute very tight rigorous bounds on the auto-convolution constant $${\nu }_2^2=\underset{f}{\inf }\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_2^2=\underset{f}{\inf }{\int }_{-1}^1(f\ast f{)}^2$$ where the infimum is taken over all unit mass functions $f\in L^1(-1\mathrm{/}2,1\mathrm{/}2)$. This quantity arises in additive combinatorics, particularly in the study of Sidon sets. Our bounds give the first 128 digits of ${\nu }_2^2$, and so substantially improve previous bounds on this quantity due to White, Green, and Martin & O’Bryant.