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We discuss three convolution inequalities that are connected to additivecombinatorics. Cloninger and the second author showed that for nonnegative $f\in L^1(-1/4, 1/4)$, $$ \max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}}{f(t-x)f(x) dx} \geq 1.28 \left( \int_{-1/4}^{1/4}{f(x) dx}\right)^2$$ which isrelated to $g-$Sidon sets (1.28 cannot be replaced by 1.52). We prove a dualstatement, related to difference bases, and show that for $f \inL^1(\mathbb{R})$, $$ \min_{0 \leq t \leq 1}\int_{\mathbb{R}}{f(x) f(x+t) dx}\leq 0.42 \|f\|_{L^1}^2,$$ where the constant 1/2 is trivial, 0.42 cannot bereplaced by 0.37. This suggests a natural conjecture about the asymptoticstructure of $g-$difference bases. Finally, we show for all functions $f \inL^1(\mathbb{R}) \cap L^2(\mathbb{R})$, $$ \int_{-\frac{1}{2}}^{\frac{1}{2}}{\int_{\mathbb{R}}{f(x) f(x+t) dx}dt} \leq 0.91 \|f\|_{L^1}\|f\|_{L^2}$$

Three Convolution Inequalities on the Real Line with Connections to Additive Combinatorics