Square-difference-free sets: the Furstenberg–Sárközy exponent

Find the growth exponent of the largest subset of {1,...,N} with no square difference, now known to lie between 0.7580758 and 1 - c.

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1	# Square-difference-free sets: the Furstenberg–Sárközy exponent
2
3	How large can a set of integers up to $N$ be if no two of its elements differ by a perfect square? The answer is a power $N^{C}$ with $C$ known only to lie between 0.7580758 and $1 - c$ for a tiny unspecified $c > 0$. Both bounds moved in 2026, and lower-bound constructions now come with machine-checked proofs.
4
5	## Problem
6
7	Let $r(N)$ be the largest size of a set $A \subseteq \{1, \dots, N\}$ such that $a - b$ is never a nonzero perfect square for $a, b \in A$. Let $C$ be the least exponent with
8
9	$$r(N) \le N^{C + o(1)} \quad (N \to \infty).$$
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11	Determine $C$, or narrow the known range. (Whether $\log r(N) / \log N$ has a limit is itself open.)
12
13	## Background
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15	Furstenberg (1977, using ergodic theory) and Sárközy (1978) proved independently that $r(N) = o(N)$. Taking multiples of a prime near $\sqrt{N}$ gives $r(N) \ge N^{1/2 - o(1)}$, and Erdős conjectured this was sharp up to polylogarithmic factors; Sárközy and then Ruzsa (1984) disproved that. Ruzsa's construction builds the set digit by digit in a suitable base, and he remarked that $3/4$ looked like the natural barrier of the method. Upper bounds improved many times (Pintz, Steiger and Szemerédi; Bloom and Maynard; Green and Sawhney) but stayed of the form $N^{1 - o(1)}$ until 2026.
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17	## What is known
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19	- Upper bound: $C \le 1 - c$ for an absolute constant $c > 0$, the first power saving ([OpenAI, preprint dated September 2026, not refereed or formalized](https://github.com/openai/math/tree/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026)); the argument gives no useful numerical value of $c$.
20	- Lower bounds: $\tfrac12$ (folklore); $\tfrac12\left(1 + \tfrac{\log 7}{\log 65}\right) \approx 0.733077$ (Ruzsa, 1984); $\tfrac12\left(1 + \tfrac{\log 12}{\log 205}\right) \approx 0.733412$ (Beigel and Gasarch, 2008, and independently Lewko, 2015).
21	- 2026 constructions: $0.752796$ (Krachun, August), the first beyond $3/4$, gluing "Paley chains" modulo several primes by the Chinese remainder theorem; then $0.753742$ (Jones), $0.75806746$ (Naslund), $0.75806770413$ (Jones) and the current record $0.7580758$ (Naslund, October), each of these four with a Lean 4 formalization of the bound. All five are preprints or code repositories, not yet refereed. Naslund remarks that this is close to the limit of the current approach.
22
23	## What would count as progress
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25	- A construction with exponent above $0.7580758$: the finite components (residue sets, chains, digit codes) and the arithmetic that turns them into an exponent. Checking the components is a finite computation over residues; the recent records also come with Lean proofs, which is the standard to aim for.
26	- A new kind of construction, since the current method appears to be near saturation.
27	- An upper bound with an explicit, reasonable value of $c$, or any proof of $C \le 1 - c$ with $c$ that can be computed.
28
29	## References
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31	1. A. Sárközy, "On difference sets of sequences of integers. I", Acta Mathematica Academiae Scientiarum Hungaricae 31 (1978), 125–149. [doi:10.1007/BF01896079](https://doi.org/10.1007/BF01896079)
32	2. I. Z. Ruzsa, "Difference sets without squares", Periodica Mathematica Hungarica 15 (1984), 205–209. [doi:10.1007/BF02454169](https://doi.org/10.1007/BF02454169)
33	3. M. Lewko, "An improved lower bound related to the Furstenberg–Sárközy theorem", Electronic Journal of Combinatorics 22 (2015), P1.32. [doi:10.37236/4656](https://doi.org/10.37236/4656)
34	4. B. Green and M. Sawhney, "New bounds for the Furstenberg–Sárközy theorem" (2024). [arXiv:2411.17448](https://arxiv.org/abs/2411.17448)
35	5. D. Krachun, "Square-difference-free sets beyond the three-quarter barrier" (2026). [arXiv:2608.01325](https://arxiv.org/abs/2608.01325)
36
37	## Source
38
39	Imported from [Optimization Constants in Mathematics, constant 4b](https://teorth.github.io/optimizationproblems/constants/4b.html), curated by Terence Tao and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.