big list - Examples of common false beliefs in mathematics

False statement: If $A$ and $B$ are subsets of $mathbb{R}^d$, then their Hausdorff dimension $dim_H$ satisfies

$$dim_H(A times B) = dim_H(A) + dim_H(B).
$$

EDIT: To answer Benoit's question, I do not know about a simple counterexample for $d = 1$, but here is the usual one (taken from Falconer's "The Geometry of Fractal Sets"):

Let $(m_i)$ be a sequence of rapidly increasing integers (say $m_{i+1} > m_i^i$). Let $A subset [0,1]$ denote the numbers with a zero in the $r^{th}$ decimal place if $m_j + 1 leq r leq m_{j+1}$ and $j$ is odd. Let $B subset [0,1]$ denote the numbers with a zero in the $r^{th}$ decimal place if $m_{j} + 1 leq r leq m_{j+1}$ and $j$ is even. Then $dim_H(A) = dim_B(A) = 0$. To see this, you can cover $A$, for example, by $10^k$ covers of length $10^{- m_{2j}}$, where $k = (m_1 - m_0) + (m_3 - m_2) + dots + (m_{2j - 1} - m_{2j - 2})$.

Furthermore, if $mathcal{H}^1$ denotes the Hausdorff $1$-dimensional (metric) outer measure of $E$, then the result follows by showing $mathcal{H}^1(A times B) > 0$. This is accomplished by considering $u in [0,1]$ and writing $u = x + y$, where $x in A$ and $y in B$. Let $proj$ denote orthogonal projection from the plane to $L$, the line $y = x$. Then $proj(x,y)$ is the point of $L$ with distance $2^{-1/2}(x+y)$ from the origin. Thus, $proj( A times B)$ is a subinterval of $L$ of length $2^{-1/2}$. Finally, it follows:

$$
mathcal{H}^1(A times B) geq mathcal{H}^1(proj(A times B)) = 2^{-1/2} > 0.
$$