On \(\Lambda\) (p)-subsets of squares.

*(English)*Zbl 0692.43005The paper is a continuation of the author’s article \([[B_ 1]\) Acta Math. 162, 227-245 (1989; Zbl 0674.43004)], the main result of which was as follows: let \(\Phi =(\phi_ 1,...,\phi_ n)\) be an orthogonal system bounded by 1 in the sup-norm and \(2<p<\infty\). Then there is a subset S of \(\{\) 1,...,n\(\}\), \(| S| >n^{2/p}\) satisfying \(\| \sum_{i\in S}a_ i\phi_ i\|_ p\leq C(p)(\sum_{i\in S}| a_ i|^ 2)^{1/2}\) for all sequences \((a_ i)\) where C(p) does not depend on n. For \(\phi_ j=e^{ij.}\) this means that S is a \(\Lambda\) (p)-set with a constant \(K_ p(S)\leq C(p)\). These notations are extended to an arbitrary \(\Phi\). From a theorem on Hilbert subspaces of \(L^ p(G)\) it follows that the exponent 2/p is best possible. Thus it can be easily deduced that there is a \(\Lambda\) (p)-set in \({\mathbb{Z}}\) which is not a \(\Lambda\) (r)-set for any \(r>p\). So is every “maximal density” set, it means such that \(\underline{\lim}_{N\to \infty}| S\cap [-N,N]| /N^{2/p}>0\). The main results of the present paper are: Theorem 1. For all \(p>4\) the set \(\{n^ 2\}^{\infty}_ 1\) contains a maximal density \(\Lambda\) (p)-set (the set \(\{n^ 2\}\) itself is not \(\Lambda\) (4)). Theorem 2. For any integer \(k\geq 1\) there is a p(k) such that for any \(p\geq p(k)\) there is a maximal density \(\Lambda\) (p)-set contained in \(\{n^ k\}\). Theorem 3. For any \(p>2\) there is a maximal density \(\Lambda\) (p)-set consisting of prime numbers (the primes themselves do not form a \(\Lambda\) (2)-set).

Using probabilistic methods and some results of \([B_ 1]\), of probabilistic type themselves, the author gives a (rather difficult and technically complicated) proof of the following result (Theorem 4): if \(2\leq q<p<\infty\) and \(\Phi\) is a finite uniformly bounded orthonormal system then there exists \(\Psi\) \(\subset \Phi\) satisfying \(K_ p(\Psi)<C(p)\) and \(| \Psi | \sim K_ q(\Phi)^{-2q/p}| \Phi |^{q/p}\). It follows that, for a subset \(S\subset \{1,...,N\}\), if \(2<q<\infty\) and \(K_ q(S)\leq C(q)| S|^{1/2}N^{-1/q}\) \((K_ q(S)\) is then called minimal) then, for \(p>q\), there is a subset \(S_ 0\) of S satisfying \(| S_ 0| \sim N^{2/p}\) and \(K_ p(S_ 0)<C(p).\)

The proofs of Theorems 1, 2 and 3 thus reduce to showing the minimality of the \(\Lambda\) (q) constants of the corresponding sets, it means for \(\{n^ 2:\) \(1\leq n\leq \sqrt{N}\}\) \((q>4)\) for Theorem 1, \(\{n^ k:\) \(1\leq n\leq N^{1/k}\}\) (q sufficiently large) for Theorem 2 and \(\{\) p: \(1\leq p\leq N\}\) (p prime, \(q>2)\) for Theorem 3. This is done in the last section by means of an estimation of some exponential sums, such as \(\sum^{N}_{1}e^{2\pi in^ 2t}\).

Using probabilistic methods and some results of \([B_ 1]\), of probabilistic type themselves, the author gives a (rather difficult and technically complicated) proof of the following result (Theorem 4): if \(2\leq q<p<\infty\) and \(\Phi\) is a finite uniformly bounded orthonormal system then there exists \(\Psi\) \(\subset \Phi\) satisfying \(K_ p(\Psi)<C(p)\) and \(| \Psi | \sim K_ q(\Phi)^{-2q/p}| \Phi |^{q/p}\). It follows that, for a subset \(S\subset \{1,...,N\}\), if \(2<q<\infty\) and \(K_ q(S)\leq C(q)| S|^{1/2}N^{-1/q}\) \((K_ q(S)\) is then called minimal) then, for \(p>q\), there is a subset \(S_ 0\) of S satisfying \(| S_ 0| \sim N^{2/p}\) and \(K_ p(S_ 0)<C(p).\)

The proofs of Theorems 1, 2 and 3 thus reduce to showing the minimality of the \(\Lambda\) (q) constants of the corresponding sets, it means for \(\{n^ 2:\) \(1\leq n\leq \sqrt{N}\}\) \((q>4)\) for Theorem 1, \(\{n^ k:\) \(1\leq n\leq N^{1/k}\}\) (q sufficiently large) for Theorem 2 and \(\{\) p: \(1\leq p\leq N\}\) (p prime, \(q>2)\) for Theorem 3. This is done in the last section by means of an estimation of some exponential sums, such as \(\sum^{N}_{1}e^{2\pi in^ 2t}\).

Reviewer: S.Hartman

##### MSC:

43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |

11L40 | Estimates on character sums |

11K06 | General theory of distribution modulo \(1\) |

43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |

42A05 | Trigonometric polynomials, inequalities, extremal problems |

11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |

42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |

##### Keywords:

orthogonal system; maximal density \(\Lambda \) (p)-set; prime numbers; uniformly bounded orthonormal system; exponential sums
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DOI

##### References:

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