My recent research has been focused on the "sum-product phenomenon" which is a vast collection of results analysing the incongruence between additive and multiplicative structure over finite sets of numbers. For instance, for any set $A$ of $N$ integers, define the sumset $2A$ and the product set $A^{(2)}$ as \[ 2A = \{a_1 + a_2 : a_1, a_2 \in A\} \ \ \text{and} \ \ A^{(2)} = \{a_1 a_2 : a_1, a_2 \in A\}. \] Just from the fact that $|A|=N$, one can deduce the trivial bounds \begin{equation} \label{triv} N \leq |2A|, |A^{(2)}| \leq N^2. \end{equation}

**Example 1.** If $A = \{1,2,\dots, N\}$, then it is clear that $2A \subseteq \{2,3, 4, \dots, 2N\}$, whence, $|2A| \leq 2N-1$. On the other hand, since $A$ contains at least $N/(10\log N)$ many primes whenever $N$ is sufficiently large (from the prime number theorem) and since product of any two primes is unique, we have that \[ |A^{(2)}| \gg\frac{N^2}{(\log N)^2}. \] Here, I use the notation $X \gg Y$ to denote that $X \geq C Y$, where $C>0$ is some constant not depending on $N$ (but it may depend on other parameters such as $s$ and $\varepsilon$).

Thus, when $A = \{1,2,\dots, N\}$, the sumset $2A$ is close to being as small as possible while the product set $A^{(2)}$ is quite close to being as large as possible. This is because the set $\{1,2,\dots,N\}$ is very additively structured but has very little multiplicative structure.

**Example 2.** One can also set $A = \{2,4,\dots, 2^N\}$, and note that $|A^{(2)}| \leq 2N-1$. Furthermore, since the sum of any two distinct powers of $2$ is unique, we see that \[ |2A| \geq \frac{N^2-N}{2}. \] In fact, in this case, the product set $A^{(2)}$ is as small as possible and the sumset $2A$ is as large as possible.

For any arbitrary set $A$ of $N$ integers, if $|A^{(2)}| \ll N$, a fundamental result in additive combinatorics called Freiman's theorem suggests that $A$ should roughly behave like a (generalised) geometric progression, see {14}. On the other hand, from the above examples, one would then expect that $|2A|$ should be very large. Perhaps with this heuristic in mind, Erdős–Szemerédi [7} conjectured that for any set $A$ of $N$ integers, either $|2A|$ is very close to $N^2$ or $|A^{(2)}|$ is very close to $N^2$. More generally, for any natural number $s$, we can define \[ sA = \{a_1 + \dots + a_s : a_1, \dots, a_s\} \ \ \text{and} \ \ A^{(s)} = \{a_1 \dots a_s : a_1, \dots, a_s \in A\}. \] With this in hand, we record the well-known sum-product conjecture as proposed by Erdős–Szemerédi.

**Conjecture 1.** For any set $A$ of $N$ integers and any $s \in \mathbb{N}$ and any $\varepsilon >0$, one has \[ \max\{|sA|, |A^{(s)}|\} \gg N^{s - \varepsilon}. \]

Erdős–Szemerédi further conjectured that this should hold true when $A$ is a set of real numbers. There have been many results towards this problem and its analogues in other rings. When $s=2$ and $A \subset \mathbb{R}$, some of the highlights have been work of Elekes {6}, who showed that \[ \max\{|2A|, |A^{(2)}|\} \gg N^{5/4}\] by reducing this to a geometric problem about incidences between points and lines. The next big advance came from work of Solymosi {15}, who used some further inherent geometric structure in this problem to replace $5/4$ with $4/3 - \varepsilon$ for any $\varepsilon >0$. The current best known result here is due to Rudnev-Stevens {13} who were able to replace $4/3 - \varepsilon$ with $4/3 + c$, for some $c \approx 1/600$.

The analogous problem, when $s$ is somewhat large (but still super small compared to $N$) has seen much less progress. A breakthrough result in this direction was due to Bourgain-Chang {3} who showed that for any set $A \subset \mathbb{Z}$ with $|A| = N$, one has \begin{equation} \label{bc} \max\{|sA|, |A^{(s)}|\} \gg N^{c (\log s)^{1/4}}, \end{equation} for some constant $c>0$. The novelty of their result is that the exponent $(\log s)^{1/4} \to \infty$ as $s \to \infty$. Their proof involved very intricate harmonic analysis combined with unique prime factorisation properties over integers. This was further simplified and improved by work of Pálvölgyi-Zhelezov {12}, who proved that \begin{equation} \label{pz} \max\{|sA|, |A^{(s)}|\} \gg N^{c \log s/\log \log s}. \end{equation} This remains the best known estimate for large $s$.

One can also analyse other notions of additive and multiplicative structure over integers. Thus, for any finite set $A \subset \mathbb{Z}$ and any $s \in \mathbb{N}$, we define $E_s(A)$ to count all $a_1, \dots, a_{2s} \in A$ such that \[ a_1 + \dots + a_s = a_{s+1} + \dots + a_{2s} . \] Let $M_s(A)$ be the the number of all $a_1, \dots, a_{2s} \in A$ such that $a_1 \dots a_s = a_{s+1}\dots a_{2s}$. For any set $A$ of $N$ non-zero integers, one trivially has \[ N^s \leq E_s(A), M_s(A) \leq N^{2s-1}. \]

**Example 3.** As before, consider the case when $A = \{1,2,\dots, N\}$. Applying Cauchy-Schwarz inequality, we get that \begin{equation} \label{eq1} E_s(A) \geq \frac{N^{2s}}{|sA|} \gg N^{2s-1}, \end{equation} which is close to the trivial upper bound above. Showing that \[ M_s(A) \ll N^{s + \varepsilon} \] holds for every $\varepsilon >0$ is slightly harder than just analysing the case of primes (as in Example 1), but this can be done via standard estimates on the divisor function. One can similarly prove that if $A = \{2,4,\dots, 2^N\}$, then \begin{equation} \label{eq2} M_s(A) \geq \frac{N^{2s}}{|A^{(s)}|} \gg N^{2s - 1} \ \ \text{and} \ \ E_s(A) \ll N^s. \end{equation}

Noting these along with Conjecture 1, one might naively conjecture that given any set finite set $A \subset \mathbb{Z}$, one has $\min\{E_s(A), M_s(A)\} \ll N^{s + \varepsilon}$ for all $\varepsilon >0$. This fails in a straightforward manner, as can be seen by setting $A = \{1,2,\dots, N\} \cup \{2,4,\dots, 2^N\}$, wherein one has \[ E_s(A) \geq E_s(\{1,2,\dots, N\}) \gg N^{2s - 1} \ \ \text{and} \ \ M_s(A) \geq M_s(\{2,4,\dots, 2^N\}) \gg N^{2s-1}. \] Recently, Balog-Wooley {1} proved that these sets are the only counter examples, that is, any finite set $A \subset \mathbb{R}$ may be partitioned as $A = B \cup C$, where \[ E_s(B) \ll N^{2s - 1 - \delta} \ \ \text{and} \ \ M_s(C) \ll N^{2s - 1 -\delta} \] for any $\delta < 2/33$. Such results have found interesting applications, including providing improvements to the original exponent in Conjecture 1. Noting Conjecture 1, Balog-Wooley put forth the natural question about whether $\delta \to \infty$ as $s \to \infty$. In a sequence of two papers {10, 11}, I was able to confirm this speculation and quantitatively strengthen it.

**Theorem 2.** Given $s \in \mathbb{N}$ and some set $A \subset \mathbb{Z}$ of $N$ integers, we can partition $A = B \cup C$ such that \[ E_s(B) \ll N^{2s - \eta} \ \ \text{and} \ \ M_s(B) \ll N^{2s - \eta} \] where $\eta \geq c\log s/ \log \log s$, with $c>0$ being some absolute constant.

Not only does Theorem 2 recover Pálvölgyi-Zhelezov's result in a straightforward manner, it has delivered further applications beyond the standard sumset-product set estimate, including to a problem of finding large additive and multiplicative Sidon sets in arbitrary sets of integers {9}. Another interesting aspect of this setting is that unlike Conjecture 1, one can never expect to have $\eta = s -\varepsilon$ in Theorem 2. In fact, Balog--Wooley {1} proved that for any $m \geq 1$, if we set \[ A = \{3,5,\dots, 2m^2-1\} \cdot \{2,4, \dots, 2^{m}\},\] then for any $B \subseteq A$ with $|B| \geq |A|/2$, one has \[ E_s(B), M_s(B) \gg |A|^{s + \frac{s-1}{3}}. \] Thus, one can never expect to have $\eta > (2s+1)/3$.

The proof of Theorem 2 relies on a variety of tools from additive combinatorics and number theory, including an analysis of when finite subsets of $\mathbb{Z}^d$ have their sumset satisfy Brunn-Minkowski type sumset growth. More recently, Gowers-Green-Manners-Tao proved a breakthrough result on the latter type of problem, which further allowed them to prove a Bourgain-Chang type sumset-product set estimate for sets $A \subset \mathbb{R}$, see {8}.

We conclude this note by mentioning that such problems have also been studied in settings such as $\mathbb{F}_p$ and a more "continuous" setup in $\mathbb{R}$. The former has provided many interesting applications to bounds on exponential sums {4} and incidence geometric problems in finite fields {5}, while the latter performed a key role in Bourgain's proof of the Erdős-Volkmann ring conjecture {2}.

Akshat Mudgal is a Postdoctoral Research Associate in Oxford Mathematics.

[1] A. Balog, T. D. Wooley, *A low-energy decomposition theorem*, Q. J. Math. 68 (2017), no.1, 207–226.

[2] J. Bourgain, *On the Erdős–Volkmann and Katz–Tao ring conjectures*, Geom. Funct. Anal. 13 (2003), no. 2, 334–365.

[3] J. Bourgain, M.-C. Chang, *On the size of k -fold sum and product sets of integers*, J. Amer. Math. Soc. 17 (2004), no. 2, 473–497.

[4] J. Bourgain, A. A. Glibichuk, S. V. Konyagin, *Estimates for the number of sums and products and for exponential sums in fields of prime order*, J. London Math. Soc. (2) 73 (2006), no.2, 380–398.

[5] J. Bourgain, N. Katz, T. Tao, *A sum-product estimate in finite fields, and applications*, Geom. Funct. Anal. 14 (2004), no. 1, 27–57.

[6] G. Elekes, *On the number of sums and products*, Acta Arith. 81 (1997), no.4, 365–367.

[7] P. Erdős and E. Szemerédi, *On sums and products of integers*, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218.

[8] W. T. Gowers, B. Green, F. Manners, T. Tao, *On a conjecture of Marton*, arXiv:2311.05762.

[9] Y. Jing, A. Mudgal, *Finding large additive and multiplicative Sidon sets in sets of integer**s*, Math. Ann. (2024), arXiv:2203.13174.

[10] A. Mudgal, Energy estimates in sum-product and convexity problems, Amer. J. Math. (2024), arXiv:2109.04932.

[11] A. Mudgal, *Unbounded expansion of polynomials and products*, Math. Ann. (2023), arXiv:2303.15910.

[12] D. Pálvölgyi, D. Zhelezov, *Query complexity and the polynomial Freiman-Ruzsa conjecture*, Adv. Math. 392 (2021), Paper No. 108043, 18 pp.

[13] M. Rudnev, S. Stevens, *An update on the sum-product problem*, Math. Proc. Cambridge Philos. Soc. 173 (2022), no.2, 411–430.

[14] T. Sanders, *The structure theory of set addition revisited**, Bull*. Amer. Math. Soc. (N.S.) 50 (2013), no. 1, 93–127.

[15] J. Solymosi, *Bounding multiplicative energy by the sumset*, Adv. Math. 222 (2009), no.2, 402–408.