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On solution‐free sets for simultaneous quadratic and linear equations
Author(s) -
Smith Matthew L.
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn073
Subject(s) - mathematics , quadratic equation , constant (computer programming) , diagonal , invariant (physics) , linear equation , integer (computer science) , natural number , combinatorics , interval (graph theory) , dilation (metric space) , mathematical analysis , mathematical physics , geometry , computer science , programming language
We consider a translation and dilation invariant system consisting of a diagonal quadratic equation and a linear equation with integer coefficients in s variables, where s ⩾ 9. We show via the Hardy–Littlewood circle method that, if a subset of the natural numbers restricted to the interval [1, N ] satisfies Gowers' definition of quadratic uniformity, then it furnishes roughly the expected number of simultaneous solutions to the given equations. If furnishes no non‐trivial solutions to the given system, then we show that the number of elements in ∩ [1, N ] grows no faster than a constant multiple of N /(log log N ) − c as N → ∞, where c > 0 is an absolute constant. In particular, we show that the density of in [1, N ] tends to 0 as N tends to infinity.