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A generalization of the classical circle problem. (English) Zbl 1292.11113

Let \(A=\{a_1\leq a_2\leq\dots\}\) be an infinite sequence of nonnegative integers tending to infinity. For a real number \(x\geq0,\) let \(r_k(A,x)\) denote the number of solutions \((a_{i_1},\dots,a_{i_k})\in A^k\) of \[ a_{i_1}+\cdots+a_{i_k}=n~. \] For a positive constant \(c,\) let \[ P_k(A,c,x)=r_k(A,x)-cx~. \]
The case \(k=2\) of the preceding notations is related
(1) to the Erdős-Fuchs theorem; its Jurkat-Montgomery-Vaughan version [H. L. Montgomery and R. C. Vaughan, A tribute to Paul Erdös, 331–338 (1990; Zbl 0715.11005)] states that \(r_2(A,x)=cx+o(x^{1/4})\) cannot hold for any constant \(c>0~;\)
(2) to the classical circle problem in which \(A=\{0^2,(-1)^2, 1^2, (-2)^2, 2^2,\dots\}~\) and the problem is to evaluate \(P(x):=P_2(A,\pi,x)~.\)
Since it is well known that \[ \int_0^X P^2(x)~\,dx=CX^{3/2} +Q(X) \] (\(C\) is approximatively 1.68396; see [Y.-K. Lau and K.-M. Tsang, Math. Proc. Camb. Philos. Soc. 146, No. 2, 277–287 (2009; Zbl 1229.11125)] and W. G. Nowak [Acta Arith. 113, No. 3, 259–272 (2004; Zbl 1092.11039)] for bounds on \(Q(X)\)), the authors propose to study the following question.
“Is it true that for any sequence \(A\) as above, any \(c>0\) and any \(\varepsilon>0~,\) there is a constant \(C=C(A,c,\varepsilon)\) such that, for sufficiently large \(X~,\) \[ \int_0^X P_2^2(A,c,x)\,dx\geq CX^{(3/2)-\varepsilon}~?~" \]
The authors prove that this is true under the assumption that \(r_2(A,x)=cx+o(x)~.\) Furthermore, they prove a similar result for \(I(X):=\int_0^X P_k^2(A,c,x)\,dx\) and they conjecture that there is a constant \(C=C(A,c,k)>0\) such that, for sufficiently large \(X\), \(I(X)\geq CX^{3/2}\).
As \(A\) is arbitrary, the circle method cannot be applied. The proofs are based on detailed study of generating functions and on evaluation of sums, series sums and integrals.

MSC:

11P21 Lattice points in specified regions
11B34 Representation functions
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