Counting monochromatic solutions to diagonal Diophantine equations

Counting monochromatic solutions to diagonal Diophantine equations, Discrete Analysis 2021:14, 47 pp.

An important subfield of Ramsey theory concerns questions of the following type: for which systems of equations $E_1,\dots,E_k$ in variables $x_1,\dots,x_m$ is it the case that for every finite colouring of the positive integers there is a solution $(x_1,\dots,x_m)$ of the equations such that $x_1,\dots,x_m$ all have the same colour? Such a system of equations is called partition regular. A famous theorem of Rado gives a complete answer to this question when the equations $E_i$ are all linear. Writing the coefficients out as rows of a matrix, the columns condition is that there should be a partition of the columns into sets $C_0,C_1,\dots,C_s$ such that the columns in $C_0$ add up to 0, and for each $i>0$ the columns in $C_i$ are rational linear combinations of the columns in $C_0,\dots,C_{i-1}$. Rado proved that a system of linear equations is partition regular if and only if it satisfies the columns condition. This result simultaneously generalizes Schur’s theorem, which states that one can always find a monochromatic solution to the equation $x+y=z$ (to see this, consider the $1\times 3$ matrix $\begin{pmatrix}1&1&-1\\ \end{pmatrix}$), and van der Waerden’s theorem (if one takes the equations $a=x_1, a+d=x_2,\dots,a+(m-1)d=x_m$, it is easy to check that the corresponding matrix satisfies the condition).

This paper concerns polynomial equations of degree greater than 1. As one would expect, in this case the problem becomes much harder, and several basic questions are unsolved, a well-known example of which is whether the equation $x^2+y^2=z^2$ always has a monochromatic solution, though in the case of two colours the problem has now been solved using a SAT solver by Marijn Heule, Oliver Kullmann and Victor Marek, who showed that there is a 2-colouring of $\{1,2,\dots,7824\}$ with no monochromatic solution, but no such 2-colouring of $\{1,2,\dots,7825\}$. However, there are a number of special cases where results have been obtained, such as a result of Vitaly Bergelson that states that the equation $x-y=z^2$ is partition regular. (This is not to be confused with the statement that one can find two numbers with the same colour that differ by a perfect square: that would tell us that $x$ and $y$ have the same colour, but Bergelson’s result shows that we can require $z$ to have the same colour as well.)

Several of these results have been proved using “qualitative methods” that give either bad bounds or no bounds at all. This paper is concerned with quantitative aspects of the question, and in particular with the question of counting how many monochromatic solutions there must be. For several instances of the problem bounds are obtained that are sharp up to an absolute constant. For example, for Bergelson’s result with $r$ colours, it is shown that if the integers up to $N$ are coloured, then the number of monochromatic solutions to $x-y=z^2$ must be at least $cN^{3/2^r}$, where $c$ is a positive constant that depends on $r$ only. A simple observation shows that this bound is sharp. Indeed, in any set of the form $(m,m^2]$ there can be no solutions, so (ignoring integer parts) one can take colour classes $(N^{1/2},N], (N^{1/4},N^{1/2}]$, and so on up to $(N^{1/2^{r-2}}, N^{1/2^{r-1}}]$, each of which contains no solutions, and one colour class $\{1,2,\dots,N^{1/2^{r-1}}\}$ that has $O(N^{3/2^r})$ solutions. (To see this last estimate, note that if we seek solutions in $\{1,2,\dots,m\}$, then there are at most $\sqrt m$ possibilities for $z$ and for each $z$ there are at most $m$ possibilities for $x$.)

Aside from their intrinsic interest, one of the justifications for proving counting results of this kind is that it obviates the need to rule out certain solutions as “trivial” or “degenerate”. For example, this is a familiar aspect of the proof of Roth’s theorem, where the analytic proofs show that a dense set contains many arithmetic progressions of length 3, which implies immediately that some of the solutions are not of the trivial form $(x,x,x)$. It is also possible that these estimates will be useful as stepping stones towards more complicated results.

In general, the equations that this paper looks at are those of the form
$$a_1x_1^2+\dots+a_sx_s^2=b_1y_1+\dots+b_ty_t.$$
One of the main results comes close to characterizing which of these equations is partition regular – the only examples not covered are those of the form $a(x^2-y^2)=bz^2+cw$.

The proofs involve a combination of tools, including the circle method, a Fourier analytic transference principle, and an arithmetic regularity lemma. In addition, the author develops certain “mixed restriction estimates” as inputs. (A restriction estimate is typically a bound for an $L_p$ norm of the Fourier transform of the restriction of a function $f$ to a set, given information about $f$ and the set, though the result shown in this paper concerns more general multipliers.) Along the way the author establishes various more robust versions of well known results, including “asymmetric versions”, which are a mixture of density and colouring results: that is, some of the variables are required to belong to dense sets and others to be of the same colour.