\(\Lambda_p\)-sets in analysis: Results, problems and related aspects (Q2760171)

This survey is definitely not an introduction to \(\Lambda_p\)-sets. It describes the interplay of this notion with various domains of mathematics as number theory, partial differential equations, harmonic and functional analysis, etc., mostly due to the author. The great density of results does not allow to give a summary of the paper. The main parts are:NEWLINENEWLINENEWLINE1) the construction, for \(p>2\), of ``true'' \(\Lambda_p\)-sets, i.e., of sets which are \(\Lambda_p\) but not \(\Lambda_q\) for any \(q>p\) (sketch of proof) and Rosenthal's theorem for \(1\leq p<2\);NEWLINENEWLINENEWLINE2a) comments on Sidon sequences (\(B_2\)-sets), with results of Erdős, Ruzsa, Ajtai, Komlós and Szemeredi; NEWLINENEWLINENEWLINE2b) Sidon sets with Pisier's characterization; NEWLINENEWLINENEWLINE2c) squares in arithmetic progressions with an estimation of Bombieri, Granville and Pintz, based on effective versions of Falting's theorem, which improves the one given by Szemeredi's theorem (or even Gowers's effective version); NEWLINENEWLINENEWLINE2d) estimation \(o (m^{1/4})\) of the \(\Lambda_4\) constant of an arbitrary set of \(m\) squares by combinatorial results of Balog and Szemeredi and Freiman's theorem; it is not known whether the set of the squares is \(\Lambda_p\) for \(p<4\);NEWLINENEWLINENEWLINE3) the Kolmogorov rearrangement problem: given an orthonormal system \((\phi_n)_{n=1,2,\ldots}\), does there exist a permutation \(\pi\) of the integers such that, for every square summable sequence \((a_n)_n\), the partial sums \(\sum_{n=1}^N a_n \phi_{\pi(n)}\) converge almost everywhere? A positive solution to this problem would give Billard's theorem on the convergence almost everywhere of Walsh-Fourier series, whose analogue for the trigonometric system is Carleson's theorem; a positive answer to Kolmogorov's problem could be obtained by a positive answer to Garsia's conjecture; a sketch of proof of a partial answer to Garsia's conjecture is given;NEWLINENEWLINENEWLINE4) eigenfunctions of the Laplacian on \({\mathcal T}^d\) and estimations of the \(\Lambda_p\) constant of lattice points on spheres (on page 214, line 13, the reference would be [10] instead of [7]); quantum limits;NEWLINENEWLINENEWLINE5a) norm of the restriction to the sphere (or to a paraboloid) of the Fourier transform of functions in \(L^p({\mathcal R}^d)\); NEWLINENEWLINENEWLINE5b) application to Strichartz's inequality for the linear Schrödinger group \((\text{e}^{it\Delta})_{t\in {\mathcal R}}\) with application to the Schrödinger equation (sketch of proof); NEWLINENEWLINENEWLINE5c) for the periodic case, replacing \({\mathcal R}^d\) by \({\mathcal T}^d\), the analogue of Strichartz's inequality fails, and estimates are given to \(\|\text{e}^{it\Delta} \phi\|_{L^q({\mathcal T}^{d+1})}/\|\phi\|_2\), for \(\phi\) with \(\text{supp} \widehat \phi\) bounded; this is connected to the evaluation of the \(\Lambda_p\) constant of the set \(\{(n,|n|^2)\in {\mathcal Z}^{d+1}\); \(|n|\leq N\}\);NEWLINENEWLINENEWLINE5d) discussion on Besicovitch sets in \({\mathcal R}^d\): their Hausdorff dimension (Conjecture 6.7), the link with Stein's conjecture, which may be read as \(\|\widehat f_{|S}\|_{L^1(d\sigma)}\leq C_p\|f\|_p\) for \(p<2d/(d+1)\), and with Conjecture 6.8 on the Kakeya maximal function; partial answers are mentioned: of T. Wolff, of Tao, Vargas and Vega, and of the author, based on Gowers's proof of the Balog-Szemeredi's theorem;NEWLINENEWLINENEWLINE5e) comments on the Bochner-Riesz conjecture, which implies Stein's restriction conjecture (partial answers by Carleson and Sjölin, Hörmander, C. Fefferman and by the author);NEWLINENEWLINENEWLINE6) distribution of Dirichlet sums: some estimates on the \(L^2\) norm of partial Dirichlet sums lead to estimates of the \(\Lambda_{2k}\) constant of the set \(\{[N^k\log n] ;\;n=1,\ldots,N\}\); Montgomery's conjecture on Dirichlet sums (Conjecture 7.4) which, as mentioned by the author on page 220, implies the Conjecture 6.7 on Besicovitch sets (see Proposition 7.10); this conjecture implies the ``density hypothesis'', i.e., estimation on the number of zeros of the Riemann zeta function (Proposition 7.5), which in turn gives an upper estimation on the difference of consecutive primes; a partial answer to the density hypothesis is given by the author (the reference [7] on page 226, line 8, should be [14]); another conjecture (Conjecture 7.6) is shown to imply Conjecture 7.4; comments on the meaning of Conjecture 7.6 are then given with a sketched proof of Proposition 7.7, which is a probabilistic counterpart of Conjecture 7.6 with other consequences.NEWLINENEWLINENEWLINENeedless to say, the reading of this paper is highly recommended.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].