This is a book version of my blog, covering many (though not all) of my articles in 2007, reworked into a publishable format (and in particular, with formal and updated references).
p. 21In Proposition 1.8, $k/\varepsilon+1$ should be $k/\varepsilon+k$.
p. 22In Proposition 1.10, $2^{1/\varepsilon}+1$ should be $2^{1/\varepsilon+1}$.Page 51: In Lemma 1.34(2), $hf'(x)$ should be $hL$.
p. ?In the start of Section 1.9, the requirement $a_d \neq 0$ should be added to the definition of a linear recurrence sequence.
p. 54The phrase “nonstandard ultrafilter on ${*} {\Bbb N}$” requires some clarification. It should be “ultrafilter on ${*}{\Bbb N}$ that extends an ultraproduct of nonprincipal ultrafilters on ${\Bbb N}$“.
Section 2.2, page ???: “need thus need” should be “thus need”.
p. ?In the proof of Lemma 2.28, in the final two displays, the left absolute value and integral signs should be interchanged.
p. 86In the final display, the exponent of $p$ on the LHS should be deleted. Similarly on the first display of page 87.
p. 95Figure 1: The graph of $K_{3,3}$ is missing two edges (namely, the long diagonal edges between opposite corners of the graph).
p. 96The inequality $|E| \leq 3|V| - 6$ is only valid for connected graphs with at least one cycle. In general, one only has $|E| \leq 3|V|$. One must then delete all terms involving the number 6 in the rest of the section. One can then replace “To solve the optimisation problem exactly, one needs to solve a cubic; but we can perform a much cheaper computation” by “One can solve the optimisation problem exactly, but we can perform a cheaper computation”.
p. 101Footnote 67 is incorrect and should be deleted.
p. 102$(A+A) \times (A \cdot A)$ should be $(A \cdot A) \times (A + A)$.
p. 103The sentence “More generally, any Riemannian manifold…” is incorrect and should be deleted, replaced instead with the sentence “Similarly, the hy0erbolic plane ${\bf H}$ is isomorphic to $SL(2,{\bf R})/SO(2)$“.
p. 104In first paragraph: the modular curve should be $SO(2,{\mathbf R}) \backslash SL(2,{\mathbf R}) / SL(2,{\mathbf Z})$. In the footnote, add “and an obvious left action of $SO(2,{\mathbf R})$,” after the final comma.
p. 128In the second and third displays, $\frac{1}{t^{p(1-p)}}$ should be $t^{p(1-p)}$.
p. 129“decaying faster” should be “decaying faster or having smaller amplitude”.
p. 143The proof that $1/\alpha \leq \hbox{maxflow}$ is not correct as stated (the perturbations indicated do not preserve the property of being an $\alpha$-flow). A correct argument is as follows. Call an edge of an $\alpha$-flow unsaturated if it has weight strictly between 0 and $\alpha$, and similarly call a vertex unsaturated if its net inflow or outflow is strictly less than $\alpha$. Observe that if e is an unsaturated edge, then the final vertex u of e will either have an unsaturated edge leading out of it (if u is unsaturated) or another unsaturated edge leading into it (if u is saturated). Similarly, the initial vertex u’ of e will either have an unsaturated edge leading into it (if u is unsaturated) or another unsaturated edge leading out of it (if u is saturated). Thus, if there is at least one unsaturated edge, then by iterating the above observations, one can find an oriented cycle along unsaturated edges with the property that at any saturated vertex u, the number of edges flowing along the cycle into u equals the number of edges flowing against the cycle into u, and the number of edges flowing along the cycle out of u equals the number of edges flowing against the cycle out of u. For this cycle, one can modify the flow as indicated in the text to reduce the number of edges in the $\alpha$-flow.
p. 137In the last line of Case 1, “multiply (1.47) by…” should be “raise (1.47) to the power $r'$ and then multiply by…”, and $r+r'$ should be $(r+1)r'$.
p. 148In the first line, $\overline{F_v} = \overline{F_{-v}}$ should read $F_v = \overline{F_{-v}}$.
p. 180In the second display, $x/\xi$ should be $x \xi$.
p. 257When selecting the large prime $p$, it is necessary that $p$ is larger than 2. This is needed in order for the binomial expansion of $(1+pB)^n$ to be expressible in the desired form $\sum_{j=0}^\infty p^j P_j(n)$ where the polynomials $P_j$ have coefficients in the p-adic integers, because the number of times $p$ divides $p^k / k!$ can be bounded below by $k - k/(p-1)$, which goes to infinity as $k \to \infty$ provided that $p$ is larger than 2.
p. 267In the third display, $\sum_{d \leq 2N} \frac{\mu(d)}{d} \log^2 d$ should be $\sum_{d \leq N} \frac{\mu(d)}{d} \log^2 \frac{N}{d}$.
Section 2.9.4: The application of the tensor power trick to the Riesz-Thorin theorem is only valid in the regime $q \geq p$.
Section 2.10?: In (2.8), $\sum_d |\lambda_d|$ should be $\sum_d |c_d|$.
Contributors
Thanks to all those who have contributed corrections. Corrections received on or before 2026-07-09 were reported by the readers listed below over the years; individual per-erratum attributions for these legacy entries were not preserved in migration.