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Journal of the London Mathematical Society Advance Access originally published online on December 14, 2008
Journal of the London Mathematical Society 2009 79(2):273-293; doi:10.1112/jlms/jdn073
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© 2008 London Mathematical Society

On solution-free sets for simultaneous quadratic and linear equations

Matthew L. Smith

Department of Mathematics
University of Georgia
1023 D.W. Brooks Drive
Athens, GA 30602-7403
USA

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 A 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 A furnishes no non-trivial solutions to the given system, then we show that the number of elements in A {cap} [1, N] grows no faster than a constant multiple of N/(log log N)–c as N -> {infty}, where c > 0 is an absolute constant. In particular, we show that the density of A in [1, N] tends to 0 as N tends to infinity.

2000 Mathematics Subject Classification 11P55, 11B75, 11D09.

The author was partially supported by NSF grant DMS-010440.

Received October 8, 2007; revised September 2, 2008; published online December 14, 2008.


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