Construction techniques for some thin sets in duals of compact abelian groups
Annales de l'Institut Fourier, Volume 36 (1986) no. 3, p. 137-166

Various techniques are presented for constructing Λ (p) sets which are not Λ(p+ϵ) for all ϵ>0. The main result is that there is a Λ (4) set in the dual of any compact abelian group which is not Λ(4+ϵ) for all ϵ>0. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of Λ (p) not Λ(p+ϵ) sets, for certain values of p. The main new constructions in specific dual groups are:

– there is a Λ (2k) set which is not Λ(2k+ε) in Z(2)Z(2) for all 2k, kN and ε>0, and in Z(p)Z(p) (p a prime, p>2) for 2k<p, kN and ε>0 (answering a question in J. Lopez and K. Ross, Marcel Dekker, 1975),

– there is a Λ (2k) set which is not Λ(4k-4+ε) in Z(p ) for 2k, kN and all ϵ>0.

It is also shown that random infinite integer sequences are Λ (2k) but not Λ(2k+ϵ) for 2k, kN and ϵ>0.

Diverses techniques sont présentées pour la construction d’ensembles Δ(p) que ne sont pas Λ(p+ε) quel que soit ε>0. Il en résulte essentiellement qu’il existe un ensemble Λ(4) dans le dual de tout groupe abélien compact qui n’est pas Λ(4+ε) quel que soit ε>0. Au cours de la démonstration de nouvelles constructions sont données en groupes duaux dans lesquels des constructions d’ensembles Λ(p) et non Λ(p+ε) étaient déjà connues, pour certaines valeurs de p. Les principales nouvelles constructions en groupes duaux sont :

– il existe un ensemble Λ(2k) qui n’est pas Λ(2k+ε) en Z(2)Z(2) quel que soit 2k, kN et ε>0 ainsi que dans Z(p)Z(p) (p étant un nombre premier, p>2) pour 2k<p, kN et ε>0 (pour répondre à une question posée dans J. Lopez and K. Ross, Marcel Dekker, 1975),

– il existe un ensemble Λ(2k) qui n’est pas Λ(4k-4+ε) dans Z(p ) pour 2k, kN et tout ε>0.

Il est également démontré que des suites aléatoires illimitées en entiers sont Λ(2k) et non pas Λ(2k+ε) pour 2k, kN et ε>0.

@article{AIF_1986__36_3_137_0,
     author = {Hajela, D. J.},
     title = {Construction techniques for some thin sets in duals of compact abelian groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {36},
     number = {3},
     year = {1986},
     pages = {137-166},
     doi = {10.5802/aif.1063},
     zbl = {0586.43004},
     mrnumber = {88c:43007},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1986__36_3_137_0}
}
Hajela, D. J. Construction techniques for some thin sets in duals of compact abelian groups. Annales de l'Institut Fourier, Volume 36 (1986) no. 3, pp. 137-166. doi : 10.5802/aif.1063. http://www.numdam.org/item/AIF_1986__36_3_137_0/

[1] G. Bachelis and S. Ebenstein, On Λ(p) sets, Pacific. J. Math., 54 (1974), 35-38. | MR 52 #3887 | Zbl 0304.43013

[2] G. Benke, An Example in the Theory of Λ(p) Sets, Bollettino U.M.I., (5) 14-A (1977), 506-507. | MR 57 #7038 | Zbl 0369.42007

[3] B. Bollobas, Graph Theory, An Introductory Course, Graduate Texts in Math., Vol. 63 (1979). | MR 80j:05053 | Zbl 0411.05032

[4] A. Bonami, Etude des Coefficients de Fourier des fonctions de Lp(G), Ann. Inst. Fourier, Grenoble, 20, fasc. 2 (1970), 335-402. | Numdam | MR 44 #727 | Zbl 0195.42501

[5] R. Bose and S. Chowla, Theorems In The Additive Theory of Numbers, Comment. Math. Helv., 37 (1962/1963), 141-147. | MR 26 #2418 | Zbl 0109.03301

[6] R. Edwards, E. Hewitt and K. Ross, Lacunarity for Compact Groups I, Indiana University Math. Journal, 21 (1972), 787-806. | MR 45 #6981 | Zbl 0221.43007

[7] P. Erdos, Problems and Results in Additive Number Theory, Colloque sur la Théorie des Nombres, Bruxelles (1955), 127-137. | Zbl 0073.03102

[8] P. Erdos and A. Renyi, Additive Properties of Random Sequences of Positive Integers, Acta. Arith., 6 (1960), 83-110. | MR 22 #10970 | Zbl 0091.04401

[9] P. Erdos and J. Spencer, Probabilistic Methods in Combinatorics, Academic Press, 1974. | MR 52 #2895 | Zbl 0308.05001

[10] T. Figiel, J. Lindenstrauss and V. Milman, The dimension of Almost Spherical Sections of Convex Bodies, Acta. Math., 139 (1977), 53-94. | MR 56 #3618 | Zbl 0375.52002

[11] C. Graham and O.C. Mcgehee, Essays in Commutative Harmonic Analysis, Springer-Verlag, 1979. | MR 81d:43001 | Zbl 0439.43001

[12] R. Graham, B. Rothschild and J. Spencer, Ramsey Theory, Wiley Interscience, 1980. | MR 82b:05001 | Zbl 0455.05002

[13] H. Halberstam and K. Roth, Sequences, Oxford University Press, 1966. | MR 35 #1565 | Zbl 0141.04405

[14] G. Hardy, Ramanujan, Chelsea, 1959. | Zbl 0086.26202

[15] G. Hardy and E. Wright, An Introduction to the Theory of Numbers, Oxford, 1938. | JFM 64.0093.03 | Zbl 0020.29201

[16] S. Kakutani, On the Equivalence of Infinite Product Measures, Ann. of Math., 49 (1948), 214-224. | MR 9,340e | Zbl 0030.02303

[17] J. Kolmos, M. Sulyok and E. Szemeredi, Linear Problems in Combinatorial Number Theory, Acta. Math. Sci. Hungar., 26 (1975), 113-121. | MR 51 #342 | Zbl 0303.10058

[18] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, 1977. | MR 58 #17766 | Zbl 0362.46013

[19] J. Lopez and K. Ross, Sidon Sets, Marcel Dekker, 1975. | MR 55 #13173 | Zbl 0351.43008

[20] D. Mumford, Introduction to Algebraic Geometry, Harvard Lecture Notes, 1967.

[21] J. Rotman, The Theory of Groups, Allyn - Bacon, 1973. | Zbl 0262.20001

[22] W. Rudin, Fourier Analysis on Groups, Interscience Publishers, 1962. | MR 27 #2808 | Zbl 0107.09603

[23] W. Rudin, Trigonometric Series With Gaps, J. Math. Mech., 9 (1960), 203-227. | MR 22 #6972 | Zbl 0091.05802

[24] H. Ryser, Combinatorial Mathematics, Carus Mathematical Monographs 14, Mathematical Association of America, 1963. | MR 27 #51 | Zbl 0112.24806

[25] B.L. Van Der Waerden, Modern Algebra, Ungar, 1953.

[26] J.H. Van Lint, Introduction to Coding Theory, Graduate Texts in Mathematics, 86, Springer-Verlag, 1980.

[27] A. Zygmund, Trigonometric Series, Cambridge University Press, 1959. | Zbl 0085.05601