INDEX almost periodic function balanced pair 20, Banach upper density Banach lower density compact extension conditional expectation density Hales-Jewett theorem decomposition of measure enhanced IP*-set (E-IP*-set) E-IP* + -set ^-sequence Furstenberg correspondence principle Hilbert-Schmidt operator Hindman's theorem integral polynomial form IP-convergence 12-13, IP polynomial recurrence theorem IP polynomial Szemeredi theorem IP-ring IP sequence IP-set IP*-set IP* + -set IP s -set, IP* s -set IPPSZ property Lebesgue system linearly independent IP sets 26 21, 50 1, 81 70, 81 28 25 91 25 5, 65 65 10 1, 81 27 13,22 40 22-23 3 13, 55 10 10 2-3 4 7, 65 79 55 24,63 73 lower density mild mixing Milliken-Taylor theorem mixing (relative), mixing set non-degenerate IP set (NIP-set) NIP*-set non-negative definite kernel orthogonal projection polynomially enhanced IP*-set (PE-IP*-set) PE-IP* + -set 4 8 23 28 73 73 28 23 5, 65 65 P E T induction 39-41, 43 polynomial expression polynomial form polynomial Hales-Jewett theorem 13., polynomial Szemeredi theorem primitive extension refinement rigid function self-adjoint kernel syndetic (relatively dense) Szemeredi's theorem total ergodicity total weak mixing uniform density upper density VIP-system weak mixing Y -kernel 11 40 19-20 2 28, 36 30 7, 8 27 4 1 68 8,92 7 1 90 6-7 27 106
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