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A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation $$n\in \mathcal S \iff dn\in \mathcal S$$ holds for almost all positive integers $n$. Typical examples of such sets are $$\mathcal Q_\gamma =\{n : P(n)>n^\gamma\}$$ where $P(n)$ denotes the greatest prime factor of $n$ and $0<\gamma<1$.

The lower asymptotic density of a set $\mathcal A$ is defined by $$\underline{d}(\mathcal A)=\liminf_{x\to \infty} \frac 1x \left\lvert \{n\le x : n\in \mathcal A\}\right\rvert$$ and upper density defined similarly using $\limsup$.

Hildebrand [1] proved that if a set $\mathcal A$ is stable and has positive lower density, then $$\underline{d}\left (\mathcal A\cap \left (\mathcal A+1\right)\right) >0$$ implying the fact that for every $\epsilon>0$, there are infinitely many integers $n$ such that both $n$ and $n+1$ has a prime factor $>n^{1-\epsilon}$.

This leads us to the Stable Set Conjecture :-

If a set $\mathcal A$ is stable and has positive lower density, then $$\underline{d}\left (\mathcal A\cap \left (\mathcal A+1\right)\cap \left (\mathcal A+2\right)\cdots \cap \left (\mathcal A+k\right)\right) >0$$ for any $k\in \mathbb N$.

I want to know what progress have been made on this conjecture so far. Please use appropriate citations.

[1] Hildebrand, Adolf, On a conjecture of Balog, Proc. Am. Math. Soc. 95, 517-523 (1985). ZBL0597.10056.

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There is a subsequent 1989 paper by Hildebrand, "On integer sets containing strings of consecutive integers" which shows that the if the set satisfies $d(A)>\frac{k-2}{k-1}$ then the conjecture holds for that set and that choice of $k$. I'm not aware of any further progress on the general conjecture.

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  • $\begingroup$ @SayanDutta As $k$ gets larger $(k-2)/(k-1)$ gets closer to 1, so one gets further from $1/2$ as the needed minimum. $\endgroup$
    – JoshuaZ
    Commented Nov 11, 2023 at 14:05
  • $\begingroup$ Ah sorry, my brain wasn't braining (+1 though) $\endgroup$
    – Sayan Dutta
    Commented Nov 11, 2023 at 14:12
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    $\begingroup$ In my paper arxiv.org/abs/1904.05096 with Teravainen (see Theorems 1.8-1.10) we can lower the density thresholds slightly, from $1 - \frac{1}{k-1}$ to $1 - \frac{1}{k-4/3+o(1)}$, assuming a Matomaki-Radziwill type equidistribution hypothesis on $A$. Further progress may require resolving the Fourier uniformity conjecture (related to the Chowla and Sarnak conjectures), and seems difficult with current technology. $\endgroup$
    – Terry Tao
    Commented Nov 11, 2023 at 17:01
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Seems to have been disproven, for all $k > 1$: A counterexample to Hildebrand's conjecture on stable sets, Redmond McNamara.

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