Improved sum-difference inequalities in abelian groups
For all finite subsets $X,Y$ of an abelian group, we prove \[ |X-Y|\le |X+Y|^{\lamI},\qquad \lamI=\frac{9451\e-3286}{5378\e+787}=1.454277448906\ldots, \] improving the classical exponent $3/2$. We also prove that a universal two-set exponent $\lambda$ over the integers implies an upper bound $2-1/\lambda$ for the Gyarmati--Hennecart--Ruzsa constant, the supremum $\theta^*$ of $t$ such that $|A-B|\gg|A+B|^t$ and $|A+B|\ll|A|$ for arbitrarily large $A,B\subset\Z$. Consequently, \[ \theta^*\le\frac{13524\e-7359}{9451\e-3286}=1.312373302115\ldots, \] improving their bound $4/3$. The first…
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