Added notation for int mult complexity

.gitignore

@@ -11,3 +11,4 @@
 *.blg
 *.xoj
 *.auxlock
+*.vtc

approx_alg.tex

@@ -88,6 +88,12 @@ The function $\degree(\cdot)$ takes a circuit \circuit as input and outputs the
 A subcircuit of a circuit $\circuit$ is a circuit \subcircuit such that \subcircuit is a DAG \textit{subgraph} of the DAG representing \circuit.  The sink of \subcircuit has exactly one gate \gate.
 \end{Definition}
 
+Finally, we will need the following notation for the complexity of multiplying large integers:
+\begin{Definition}[$\multc{\cdot}{\cdot}$]
+In a RAM model of word size of $W$-bits, $\multc{M}{W}$ denotes the complexity of multiplying two integers represented with $M$-bits. (We will assume that for input of size $N$, $W=O(\log{N})$.
+\end{Definition}
+We note that when doing arithmetic operations on the RAM model for input of size $N$, we have that $\multc{O(\log{N})}{O(\log{N})}=O(1)$. More generally we have $\multc{N}{O(\log{N})}=O(N\log{N}\log\log{N})$.
+
 \subsection{Our main result}
 In the subsequent subsections we will prove the following theorem.
 

macros.tex

@@ -419,7 +419,9 @@
 %%%Adding stuff below so that long chain of display equatoons can be split across pages
 \allowdisplaybreaks
 
+%Macro for mult complexity
 
+\newcommand{\multc}[2]{\overline{\mathcal{M}}\left({#1},{#2}\right)}
 
 
 %%% Local Variables:

main.vtc

@@ -1 +0,0 @@
-\contitem\title{Standard Operating Procedure in Bag PDBs Queries Considered Harmful}\author{Su Feng, Boris Glavic, Aaron Huber, Oliver Kennedy, and Atri Rudra}\page{23:1--23:45}