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I'm doing my Bachelor's Thesis on Gödel's constructible universe $L$.

I'm interested in the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the constructible universe, Marek and Srebrny, 1973. Of course, the issue is deeply linked to the fine structure of $L$, as studied in The fine structure of the constructible hierarchy, Jensen, 1971.

I'm searching for recent literature on these issues. Especially, since the gaps in $L$ itself seem to have been thoroughly studied long ago, my advisor has recommended I look for similar results for generalizations or extensions of $L$, such as $L(\mathbb{R})$ and its $\Sigma_1$-Gaps, or inner models of large cardinals, such as $L(U)$. I haven't found anything yet, other than Steel's Scales in $L(\mathbb{R})$, and I'd be very grateful for any information or advice you might have.

The standard text, Constructibility by Devlin, doesn't seem to touch upon gaps of this kind. Gap-$n$ morasses are studied, but from what I understood they're an unrelated concept. I'm also aware of modern developments of fine structure theory, such as hyperfine structure, but these seem again to touch only upon Gap-$n$ morasses.

Thanks in advance :)

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    $\begingroup$ Steel has a paper on gaps in $K(\mathbb R)$ and a few related papers in the reissued Cabal vol. I and a follow up in the JSL. $\endgroup$
    – Andrés E. Caicedo
    Commented Mar 14, 2022 at 22:08
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    $\begingroup$ For posterity: the OP's thesis is now uploaded to the arXiv, "The real numbers in inner models of set theory", also with a brief mention of the concept of higher-order gaps in MO answer #424952. $\endgroup$
    – C7X
    Commented Jan 19, 2023 at 12:37

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