The Theorems Of Sárk?zy Type In Function Fields

Let A(?)N+={1.2.3...}.Lovasz posed the following conjecture:If(?)|A∩[1,N]|/N>0,then A-A={a-a’:a,∈ A} must contain squares of non-zero nature numbers,where let|A∩[1,N| denote the cardinality of A ∩[1,N].In 1978,to prove Lovasz’s conjecture Sarkozy got the following result.Let N ∈N with N≥ 2 and A(?)N ∈[1.N].If n2(?)A-A,all n ∈N+,then|A|≤C1N((ln ln N)2/ln N)1/3 where C1>0 is an absolute constant.The main aim of this paper is to study the analogue of the above result in function fields.Let q ∈ N+and let Fq be the finite field of q elements.Let p be the characteristic of Fq.Let A=Fq[t]be the polynomial ring over Fq.Let K=Fg(t)denote the field of fractions of A.For N∈N+,define GN={m∈A:deg m<N}.In 2013.Le and Liu first obtained analogue the following of Sarkozy’s theorem in the function field K.Let N∈N with N≥ 2 and A(?)Gn.Ifp>2 and m2(?)A-A,all m ∈ A\{0},then|A|≤C4qN(ln N)7/N,where C4>0 depending only on q.In this paper,result of Le and Liu is generalized:Let k,N∈N with k,N>2 and A(?)GN.If p(?)k and mk(?)A-A,all m A\{0},then|A|≤C5qN(ln N/N)1/k-1,where C5>0 depending only on k and q.It is easy to see that p(?)2(?)p>2 and In N<(ln N)7 when N≥ 3.Hence,in the special case when k=2.the above theorem also improves the results of Le and Liu if constants C5 and C4 are ignored.