Suppose $K$ is a subset of $[0,1]$ with the following property: for almost $x,y \in K$, we have

$$\frac{x+y}{2} \not\in K.$$

(Here, "almost in $K$" means "in $K$ except for a countable subset").

Such a set must have holes, and the Cantor tridiagonal set has this property. What can we say about the Hausdorff dimension of $K$? Is it true that it is less than $\ln 3 / \ln 2$?

Can we even say that the Hausdorff measure $H^{\ln 3/ \ln 2}(K)$ is finite?