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On $$\Lambda$$ (p)-subsets of squares. (English) Zbl 0692.43005
The 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 $$\sum^{N}_{1}e^{2\pi in^ 2t}$$.
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
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##### References:
 [1] G. F. Bachelis and S. E. Ebenstein,On $$\Lambda$$(p)-sets, Pacific J. Math.54 (1974), 35–38. [2] G. Bennett, L. Dor, V. Goodman, W. B. Johnson and C. Newman,On uncomplemented subspaces of L p, 1
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