Rank Amplification for Shifted Equal Values of Euler's Totient Function

Let $S_h^{\phi }(x)$ denote the number of integers $n\le x$ for which $\phi (n)=\phi (n+h)$. For the unit shift, we prove $S_1^{\phi }(x)\ll x\exp -(1\mathrm{/}2-o(1))\sqrt{\log x,{\log }_{2}x}$. More generally, put $A={\log }_{3}x+{\log }_{4}x-\log 2$, $G=\sqrt{\log x,A}$, and $V=\log x\mathrm{/}G$. For every fixed integer $J\ge 1$, uniformly for $1\le h\le \exp G\mathrm{/}\sqrt{J}$, we obtain $S_h^{\phi }(x)=D_{h,>Y_J}^{\phi }(x)+O_J(x\exp -\sqrt{J},G+o_J(V))$, where $Y_J=\exp \sqrt{J},G$. Here $D_{h,>Y_J}^{\phi }(x)$ is the above-cutoff part of the classical Graham–Holt–Pomerance same-support family; it is empty for odd $h$. A moving choice $J\asymp {\log }_{2}x\mathrm{/}{\log }_{3}x$ gives the unit-shift estimate and an analogous decomposition for a uniform range of shifts. The proof combines the smooth-totient theorem of Banks–Friedlander–Pomerance–Shparlinski with labelled supplier systems, a shifted divisor convolution, and an injective encoding of large supplier products into weighted friable tuples.