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Klaus Roth, who made fundamental contributions to analytic number theory, died this Tuesday, aged 90.

I never met or communicated with Roth personally, but was certainly influenced by his work; he wrote relatively few papers, but they tended to have outsized impact. For instance, he was one of the key people (together with Bombieri) to work on simplifying and generalising the large sieve, taking it from the technically formidable original formulation of Linnik and Rényi to the clean and general almost orthogonality principle that we have today (discussed for instance in these lecture notes of mine). The paper of Roth that had the most impact on my own personal work was his three-page paper proving what is now known as Roth’s theorem on arithmetic progressions:

Theorem 1 (Roth’s theorem on arithmetic progressions) Let ${A}$ be a set of natural numbers of positive upper density (thus ${\limsup_{N \rightarrow\infty} |A \cap \{1,\dots,N\}|/N > 0}$). Then ${A}$ contains infinitely many arithmetic progressions ${a,a+r,a+2r}$ of length three (with ${r}$ non-zero of course).

At the heart of Roth’s elegant argument was the following (surprising at the time) dichotomy: if ${A}$ had some moderately large density within some arithmetic progression ${P}$, either one could use Fourier-analytic methods to detect the presence of an arithmetic progression of length three inside ${A \cap P}$, or else one could locate a long subprogression ${P'}$ of ${P}$ on which ${A}$ had increased density. Iterating this dichotomy by an argument now known as the density increment argument, one eventually obtains Roth’s theorem, no matter which side of the dichotomy actually holds. This argument (and the many descendants of it), based on various “dichotomies between structure and randomness”, became essential in many other results of this type, most famously perhaps in Szemerédi’s proof of his celebrated theorem on arithmetic progressions that generalised Roth’s theorem to progressions of arbitrary length. More recently, my recent work on the Chowla and Elliott conjectures that was a crucial component of the solution of the Erdös discrepancy problem, relies on an entropy decrement argument which was directly inspired by the density increment argument of Roth.

The Erdös discrepancy problem also is connected with another well known theorem of Roth:

Theorem 2 (Roth’s discrepancy theorem for arithmetic progressions) Let ${f(1),\dots,f(n)}$ be a sequence in ${\{-1,+1\}}$. Then there exists an arithmetic progression ${a+r, a+2r, \dots, a+kr}$ in ${\{1,\dots,n\}}$ with ${r}$ positive such that

$\displaystyle |\sum_{j=1}^k f(a+jr)| \geq c n^{1/4}$

for an absolute constant ${c>0}$.

In fact, Roth proved a stronger estimate regarding mean square discrepancy, which I am not writing down here; as with the Roth theorem in arithmetic progressions, his proof was short and Fourier-analytic in nature (although non-Fourier-analytic proofs have since been found, for instance the semidefinite programming proof of Lovasz). The exponent ${1/4}$ is known to be sharp (a result of Matousek and Spencer).

As a particular corollary of the above theorem, for an infinite sequence ${f(1), f(2), \dots}$ of signs, the sums ${|\sum_{j=1}^k f(a+jr)|}$ are unbounded in ${a,r,k}$. The Erdös discrepancy problem asks whether the same statement holds when ${a}$ is restricted to be zero. (Roth also established discrepancy theorems for other sets, such as rectangles, which will not be discussed here.)

Finally, one has to mention Roth’s most famous result, cited for instance in his Fields medal citation:

Theorem 3 (Roth’s theorem on Diophantine approximation) Let ${\alpha}$ be an irrational algebraic number. Then for any ${\varepsilon > 0}$ there is a quantity ${c_{\alpha,\varepsilon}}$ such that

$\displaystyle |\alpha - \frac{a}{q}| > \frac{c_{\alpha,\varepsilon}}{q^{2+\varepsilon}}.$

From the Dirichlet approximation theorem (or from the theory of continued fractions) we know that the exponent ${2+\varepsilon}$ in the denominator cannot be reduced to ${2}$ or below. A classical and easy theorem of Liouville gives the claim with the exponent ${2+\varepsilon}$ replaced by the degree of the algebraic number ${\alpha}$; work of Thue and Siegel reduced this exponent, but Roth was the one who obtained the near-optimal result. An important point is that the constant ${c_{\alpha,\varepsilon}}$ is ineffective – it is a major open problem in Diophantine approximation to produce any bound significantly stronger than Liouville’s theorem with effective constants. This is because the proof of Roth’s theorem does not exclude any single rational ${a/q}$ from being close to ${\alpha}$, but instead very ingeniously shows that one cannot have two different rationals ${a/q}$, ${a'/q'}$ that are unusually close to ${\alpha}$, even when the denominators ${q,q'}$ are very different in size. (I refer to this sort of argument as a “dueling conspiracies” argument; they are strangely prevalent throughout analytic number theory.)